# How To Cut A Pentagon Into 4 Acute-Angled Triangles

## 84285 (Lopovok Problems), page 6

## File Description

A document from the archive “Lopovok’s tasks”, which is located in the category “abstracts”. All this is in the subject “mathematics” from the section “Student papers”, which can be found in the student file archive. Despite the direct connection of this archive with the Student, it can also be found in other sections. The archive can be found in the section “abstracts, reports and presentations”, in the subject “mathematics” in common files.

## Text of page 6 from document “84285

A convex quadrangle is inscribed in a circle *ABC**D**.* Prove that *AC B**D* = *AB SV* *VS AB.*

The two chords intersect inside the circle. Prove that the products of the segments of these chords are equal.

The two chords are mutually perpendicular. Prove that the sum of the squares of the segments of these chords is equal to the square of the diameter of the circle.

The circle goes through the top *AND* parallelogram *ABCV* and crosses straight *AB, AC, AB* in points *E* C |, r1. Prove that *AB AB \* *AB AB ^* *AC AC* (Fig. 40).

The polyline consists of 7 links, the angle between each two adjacent links is 150 °. Prove that this is a broken line;

has two links that lie on one straight line or are parallel.

Are given *n 2* points, not all of which lie on the same point. Prove that you can build a simple closed polyline, on the links of which all these points are placed.

The side of the square is 12 cm. Inside it is a polyline with a length of 51 cm. Prove that this polyline has at least four links.

On the sides of the triangle *ABC* outside it are squares with centers *0 \, Och,* Oz. Points *Ao, Vo, So.* midpoints of triangle sides *ABC, SCAOODV.* parallelogram (fig. 41). Prove that broken lines *0 \ ВСоОз* and *О ^ ВцСоВ* are equal.

Using the result of Problem 52, prove that the segments *0 \ 0y.* and *O ^ B* are equal and mutually perpendicular. Derive from here one of the ways of constructing a triangle by the centers of the squares built on its sides outside the triangle.

A closed polyline consists of 1989 links and does not have self-intersections. Prove that a straight line that does not pass through any vertex of a polyline does not intersect all the links of this polyline.

The tourist moved along the broken line, all the links of which had the same length, and wrote down the turns he made at its vertices: 15 ° to the right, 30 °, 90 °, 105 °, to the left 120 °, to the right 75 °, 30 °, 90 °. Was his route closed?

Polygon

A convex polygon has 1000 vertices and 2000 points are given inside it. Among those 3000 points (vertices and data), no three are collinear. The polygon is divided into triangles, the vertices of which are only the points mentioned. In this case, the triangles do not overlap and each of the 3000 points is the vertex of at least one triangle. Determine the total number of triangles.

Prove that a convex polygon has a diagonal that is greater than at least two of its sides.

What is the largest number of right angles among the interior corners of a convex polygon?

Each side of the n-gon is the diameter of the circle. Knowing that these circles contain all the interior points of the polygon, determine the possible values *P.*

Prove that a plate in the shape of a convex pentagon can be cut into three trapezoids.

Prove that the convex r-gon (n

4) can be divided into *n. 2* trapeze.

Diagonal divides convex pentagon into rhombus *ABVE* and an equilateral triangle *ВСВ.* Find the corner *ACE.*

Prove that it is possible to construct a pentagon whose sides are equal to the diagonals of some pentagon.

Construct a pentagon based on the position of the midpoints of all its sides.

Construct a pentagon based on the position of the midpoints of all its diagonals.

*AVSVER.* hexagon, the midpoints of which *K, b, M, K, O, P.* Prove that the centers of mass of the triangles *KMO* and *SHR* match.

A convex heptagon is inscribed in the circle, in which the degree measures of the three angles are equal to 120 °. Prove that among the sides of this heptagon there are two equal.

The sides of the triangle are 5, 6, 10 cm. Three straight lines, respectively parallel to the sides of the triangle, intersect in pairs outside the triangle. These straight lines intersect the sides of the triangle so that an equilateral hexagon is formed. Find its perimeter.

All angles of a convex hexagon are equal. Prove that the differences in the lengths of its parallel sides are the same.

Rectangle area

The smaller of the lateral sides of a rectangular trapezoid *and.* The other side is equal to the sum of the bases. Find the area of a rectangle whose sides are equal to the bases of the named trapezoid.

The diagonals of the rhombus are 30 and 40 cm.The circle inscribed in the rhombus touches its sides at points *A, B, C, B.* Find the area of a quadrilateral *ABCV.*

The lengths of the sides of the rectangle *and* and *B.* How to cut it into two parts, from which you can fold a square if:

a) o = 8 cm ” *B* = 18 cm; b) about 9 cm, *B* = 16 cm?

The lengths of the sides of a rectangle are expressed in whole numbers in centimeters, with the perimeter (in centimeters) and area (in square centimeters) expressed in the same numbers. Find the area of a rectangle.

Distances of the inner point M from the three vertices of the square *ABCV* such: *MA * 7 cm, *MV* = 17 cm, *MC * 23 cm.Find the area of a square.

Three parallel straight lines are given, the middle of which is at a distance from the other two by o and b. Find the area of a square whose three vertices are on these lines.

A perimeter rectangle P is inscribed in a circle of radius D. Find the area of the rectangle.

Parallelogram area

Find the area of a parallelogram along its perimeter *R* and two heights. *H \* and *H 6* *8.*

Find the angles of a triangle with 8 (o 2 b 2).

Prove that in each triangle

Is it true that in a triangle with sides a, b, c and heights *Na, b.u, Hs: (a* B c)

The lengths of the two sides of the triangle a and b, the bisectors of the angles at the third side intersect at an angle of 15 °. Find the area of a triangle.

Two equal rectangles have a common diagonal, prove that the area of their common part is more than half the area of each rectangle.

Prove that the area of a quadrangle is at most the product of the half-sums of the lengths of the opposite sides.

About a square *ABCV* a circle is described. Find a point on it *M,* so that the work *MA MV MS MV* had the greatest possible value.

Parallelogram area *ABCV* is equal to O. Vershina *M* parallelogram *AMKV* divides *Sun* so that *VM: MS* = 3: 5. Find the area of the common part of parallelograms.

Quadrangle area 2.

Three straight lines are parallel. The middle one is 4 and 7 cm away from the other two.Find the area of an equilateral triangle whose vertices lie on these three lines.

Trapezium area

The triangle is divided into three trapezoids, the common vertex of which is the center of mass of the triangle. Compare the areas of the named trapezoids.

The area of a square built on the diagonal of an isosceles trapezoid is 4 times the area of the trapezoid. Find the angle between the diagonals of the trapezoid.

The sum of the areas of the squares built on the diagonals of the trapezoid is 4 times the area of the trapezoid. Prove that the diagonals of this trapezoid are mutually perpendicular.

Trapezium bases *Sun* and *AB,* diagonals intersect at point O. Area of triangles *AVO* and *WZO* are 50 and 20 cm 2. Find the area of the trapezoid.

The angle between the diagonals of an isosceles trapezoid is 60 ° (two cases). How to cut this trapezoid into as few pieces as possible to make an equilateral triangle?

The diagonals of an isosceles trapezoid are mutually perpendicular. Side extensions *AB* and *SV* intersect at the point *M* at an angle of 30 °. Knowing that the area of a triangle *Navy* equals *ABOUT,* find the area of the trapezoid.

A trapezoid is inscribed in a semicircle with a radius of 2 cm, the perimeter of which is 10 cm.Find the area of the trapezoid.

Squares of similar figures

The area of the triangle is *8.* Each side was extended to its length in both directions. Find the area of the hexagon, which turned out when you connected the ends of the specified segments.

Into an equilateral triangle *ABC* inscribed a triangle *BEP,* whose sides are respectively perpendicular to the sides of the triangle *ABC.* Find the ratio of the areas of the triangles *BEP* and *ABC.*

Area of a triangle *ABC* equal to 120 cm 2. Each side was divided in the ratio 1: 2: 1. Three straight lines were drawn through the division points, which cut off three triangles from the triangle (Fig. 47). Determine the area of the remaining hexagon.

At the heights *VC* and *VM* diamond *ABCV* built a rhombus. Knowing that its area is half the area of a rhombus *ABCV,* find the angles of the rhombus.

The hypotenuse of a right-angled triangle is 24 cm. A straight line parallel to the smallest median divided the triangle into parts, the areas of which are 1: 7. Find the length of the line segment bounded by the sides of the triangle.

In a right-angled triangle, the two large sides of which are 8 and 10 cm, a circle is inscribed. Having constructed the tangents to it, respectively parallel to the sides of the triangle, we got a hexagon. Find its area.

The bases of the trapezoid are 7 and 17 cm. A straight line, parallel to the bases, divided the trapezoid into equal parts. Find the length of the line segment bounded by the sides of the trapezoid.

Through the interior point M of the triangle *ABC* three straight lines are drawn, respectively parallel to the sides of the triangle *ABC.* The area of the formed triangles with apex M is equal), um, 2 Y2; 2) the product of the smallest diagonal by the largest; 3) *A \ Az* X X *A \ A ^,* 4) the cube root of the doubled product of the side lengths and all diagonals outgoing from one vertex;

5) twice the product of the side by the diagonal *A \ A ^;*

6) the product of two unequal parallel diagonals. Which of these answers are correct?

The corners of the square are cut off to form a regular octagon. How much the area of the figure has decreased?

The side of a regular hexagon is *and.* A straight line is drawn through the vertex of the hexagon, dividing it into parts, the areas of which are 1: 3. Find the length of the line segment bounded by the sides of the hexagon.

Calculate the area of a polygon from the coordinates of all its vertices.

Quadrangle *ABCV* is divided into three parts by segments that do not intersect and divide the sides. *Sun* into three equal parts (fig. 50). Prove that the area of the middle part is equal to one third of the area of the quadrilateral *ABCV.*

Circumference

One circle is built on the leg of a right-angled triangle, as in the diameter. Another circle passes through the midpoints of all sides of the triangle. Under what condition are both circles equal??

Side of a square *ABCV* equal to 8cm. Find the length of the circle that goes through the points *AND* *v.* *In ty* touches the CO side of the square.

A regular dodecagon is inscribed in a circle of radius L. Its small diagonals meet at points lying on a certain circle. Determine its length.

An equilateral triangle is inscribed in a circle of radius D *ABC.* Find the length of a circle that touches a given circle and the extensions of the sides *AB* and *AS* triangle.

The radius of a circle is 2 cm. Two circles with radii of 1 cm touch each other and inwardly touch the larger circle. Find the circumference of a circle tangent to these three circles.

Perimeter of an equilateral triangle *ABC* is equal to P. Find the length of the circle that touches the side *AB* and median *AB* and *BE.*

The segment length is equal to half the circumference. There are different ways to build it:

a) Hero of Alexandria:

*b)* A. Kokhansky: *AB.* circle diameter, *SV.* tangent through point *IN; A- COB* 30 °, *SV* ZL. The sought segment. *AB* (fig. 51);

c) X. Huygens: the required segment is 8012 *IN;*

d) If the legs of a right-angled triangle are 8 and 9, then half of the circumference of the unit radius is equal to the difference between the hypotenuse and 8, 9. Check the accuracy of constructing the segment by these methods.

How are the lengths of circles, one of which is described around an equilateral triangle, and the other passes through the centers of the excircles.

Circular arc length

Chords *AB* and *SV* the circles are parallel. Prove that arcs *AS* and *BB* are equal.

Prove that if two chords of the circle are equal, then the arcs contracted by these chords are also equal.

Each side of the triangle is 6 cm. A circle with a radius of 2 cm rolls along the sides of the triangle outside of it. Determine the length of the path of the center of the circle in one revolution around the triangle.

On the side *AB a* regular hexagon *AVSVER* outside it is a square. This square moves around the hexagon so that all the time one of the vertices of the square coincides with the vertex of the hexagon. Determine the path length of the center of the square in one revolution around the hexagon.

Each vertex of a square with a side *and* is the center of a circle of radius *and.* Find the perimeter of a curved quadrangle bounded by the named circles.

The vertices of the rectangle divide the circumcircle into parts, the lengths of two of which are 1: 5. Find the radian measures of the angles that the diagonal of the rectangle makes with its sides.

Radian measures of two angles of a triangle. ^. and *-^.*

Find the ratio of the lengths of the sides of a triangle lying opposite the named angles.

Vertex *AND* equilateral triangle *ABC* is the center of the circle passing through points B and C. The bisectors of the angles *B and C* intersect the circle at points M and P. Determine the radian measure of the central angles corresponding to the arcs *RV, VS, CM, MR.*

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## Area of a circle and its parts

Perimeter of an equilateral triangle *R.* At the height of the triangle, as on the diameter, a circle is built. Determine the area of the part of the circle that is inside the triangle.

The hypotenuse of an isosceles right-angled triangle about. A circle is built on the leg, as on the diameter. Find the area of the part of the circle that is inside

*AB.* semicircle base, point *M* is on the circle. Constructed semicircles with diameters *AM* and *VM.* Prove that the sum of the areas of the holes (that is, the parts of the semicircles that are outside the large semicircle) is equal to the area

triangle *AMV.*

*AB.* the diameter of the semicircle, C is the point of this diameter ” *CO.* perpendicular to *AB,* where point *IN* is on the circle. Constructed semicircles of diameters *AC and BC* inside a semicircle. Prove that the area of a figure bounded by three named semicircles (it is called an arbelon) is equal to the area of a circle of diameter *SV* (fig. 52).

On the diameter of a semicircle *AB* equal segments are plotted *AB* and CO. On *AB* and SB, as on diameters, semicircles are built inside a large semicircle, on *Sun.* outside the large semicircle. Radius *OE* and *OR* pass through the middle *Sun* perpendicular *Sun.* Prove that the area of the figure shaded in yellow in Figure 53 is equal to the area of a circle of diameter *EP.*

Cosine theorem

Find the perimeter of a triangle whose side lengths (in centimeters) are expressed in successive odd numbers, and one of the angles is twice the sum

Calculate the angles inscribed in a circle

a quadrangle with side lengths 14, 30, 40, 48.

Prove that in a triangle *ABC: ab* soybean C

*ace* cos *IN* *Bc* coa *AND* ^. P 2.

Medians *JSC* and ID of the triangle *ABC* mutually perpendicular, prove that 5 *AB* *2* * AS* *2* *‘* *Sun* *2* *.*

Calculate *(ab* coa C ac coa B *Bc cov A): (a* *2* *B ‘* *r*

-(- *from* *2* *),* where a, b, c, */. A, /. B, /. C.* elements of one triangle.

On diameter *AB* circle point taken *M;* chord *CO* parallel *AB.* Prove that the quantity *MC* *2* *MO* *2* not

1 depends on the choice of point C.

On the sides of a triangle with side lengths 5, 6, 7, squares are built outside the triangle. Find the sum of the squares of the sides of a hexagon whose vertices are the vertices of the squares outside the triangle.

The square of the product of the lengths of the diagonals of a parallelogram is equal to the sum of the fourth powers of the lengths of two adjacent sides. Find the angles of the parallelogram.

Dot *M* is on the side *Sun* triangle *ABC.* Prove that *AM* *2* * Sun* = *AB* *2* * CM* *AS* *2* *‘VM. VS VM X X CM.*

Circles of radii 1 and 2 touch each other externally and tangent to a circle of radius 3 internally. Find the radius of the circle that touches all three named circles.

External angles of a triangle at the vertices *A, B, C* respectively *and,* (3, *at,* prove that *ab* (1. soy) *-\. ac (1.*

*–* C08 R) *BC* (1.C08 and) 4- *R* *2* *–*

Prove that in a triangle *ABC ‘.* *aa* *‘* *from*

Prove that triangle *ABC.* acute-angled if:

a) its perimeter is 17 cm, and the length of the largest side is 7 cm; b) its perimeter is 99 cm, and the length of the smallest side is 29 cm.

The center of the inscribed circle in a right-angled triangle is 7 and 5 away from the ends of the hypotenuse *-l / 2* see Find the lengths of the sides of the triangle.

Prove the correctness of formulas for calculation

area of the triangle: *8 = ^.*.\/four *a ^ b* *2*. (o 2 b 2. c 2) 2 =

=.1-.zG ^ b 2 oy *B ^ c* *2* *).* (a 4 b 4 c 4).

Prove that in a triangle *ABC:*

C08 *AND* __ __* B* C08

*IN*C C08 C G

about * B* with L

Prove that in a quadrilateral *ABCV: AB* *2* 4B 2 *Sun* *2* CO 2. 2 *AB BC. soy V. 2 sun SO owls* FROM *2АВ СО * cos *(AO).*

If the sum of the squares of the diagonals of a convex quadrilateral is equal to the sum of the squares of two opposite sides, then the extensions of the other two sides intersect along, p at right angles. Prove.

Area of a triangle *ABC* is O. Determine the value *and* *2* tue *2B* *B* *2* at *2A.*

Dot *M* is inside the triangle *ABC.* Beams *AM, VM, CM* divide the angles of the triangle into parts oi and wasp,? 1 and 3a, vi and *y-g-* Prove that *w a \ * tue) ah vi *–* em K2 X

If the rays emanating from the vertices of the triangle form with sides at these vertices such angles oi, »2, Pb

^ 2, vi “72, WHAT IS YASH OTs 8SH ?! 81n ^ 1 V ™ Y2 8P1? 2 8Sh 72, THEN THESE RAYS

intersect at one point. Prove.

Is the statement of Problem 200 true for a quadrilateral?

Prove that the bisector of the inner angle of a triangle divides the side into parts inversely proportional to the sines of the angles of the triangle adjacent to the side segments.

Heights of an acute-angled triangle *ABC* intersect at the point *H.* Prove that *AN*.-

Diagonals of a convex quadrilateral *ABCV* intersect at point O. *M \* and *Mch.* centers of mass of triangles *VOS* and *AOB, N \* and Y2. orthocenters of triangles *AVO* and *SOO.* Using the result of Problem 204, prove that the lines *M \ Mch* and Y1Y2 are mutually perpendicular.

*AB* and *AS*. chords of a circle. Continuing *AB* point marked *N* on distance *AB* from *AS* and on the continuation of the AC the point is marked *M* on distance *AS* from *AB.* Prove that *MN* is equal to the diameter of the given circle.

Sum of squares of parallelogram diagonals

Prove that in the triangle Toa. “l / 26 2 2s 2. о 2.

Using the result of problem 207, set, *what”.*

and) *t1 t1* from? =. (o 2 b 2 s 2); b) from 4 *t1 * those 4 =

Prove that in a quadrilateral the sum of the squares of the diagonals is less than the sum of the squares of the sides by the quadruple square of the distance between the midpoints of the diagonals.

Prove that a quadrilateral whose sum of squares of diagonals is equal to the sum of squares of sides is a parallelogram.

Parallelogram diagonals *ABCV* intersect at point O. Perimeters of triangles *AVO, VSO* and parallelogram, respectively 28, 30 and 48 cm.Find the diagonals of the parallelogram.

How to find the lengths of the diagonals by the lengths of the sides and the angle between the diagonals of a parallelogram?

How to find the lengths of the sides of a parallelogram from the lengths of the diagonals and the angle of a parallelogram?

TENTH CLASS

Stereometry axioms

and the consequences of them

Two points are marked on two planes. How many different planes these points define?

How many different planes can 5 points define? Support your answer by listing the planes (denoting the points with letters).

How many different planes can 5 given parallel lines define? Justify your answer by listing these planes.

The circle has a common point with each side of the quadrangle. Is it possible to assert that both of these figures lie in the same plane?

How many planes there are, each of which contains at least three vertices of a given cube?

How many areas do the planes of all cube faces split into??

On each of the three parallel edges of the cube, 2 points are marked. How many different planes can these points define??

Plane b intersects the planes a and P. Prove that if the mowing line intersects the planes, then the point of their intersection is on the straight line along which a and p intersect.

The midpoints of all the diagonals of the pentagon lie in the same plane, and no two of them coincide. Prove that all its vertices lie in the same plane.

The midpoints of all sides of a polygon with an odd number of vertices lie in the same plane. Prove that all its vertices lie in the same plane.

Are given *P * 4 points, each 4 of which lie in the same plane. Prove that all these points lie in the same plane.

Parallelism of lines in space

Prove that two lines are parallel if and only if any plane intersecting one of them intersects the other.

Points *A, B, C, B* lie outside the plane of the parallelogram *K ^ MN,* moreover *K.* middle *AB, b.* middle *BC, M*. middle of CO. Does it *N* the middle of the segment *A07*

The midpoints of the five sides of the hexagon are in the same plane. Prove that the middle of the sixth side is in the same plane.

On two intersecting planes 6 and o are given by point. How to draw straight lines through these points that do not intersect any of the named planes?

Through a straight line *I* there are two planes a and a. Two parallel lines intersect these planes: one at the points *AND* and *IN,* another. at point C and another one that you want to build.

Points *A, B, C, O* do not lie in the same plane. Prove that the midpoints of six line segments with ends at these points are the midpoints of three parallelograms.

Point M lies outside the plane of a regular hexagon *AVSOEP.* Is it true that the line passing through the midpoints of the segments *MV* and MS, parallel: a) *JSC;* b) CO?

By the condition of Problem 18, determine which sides or diagonals of the hexagon are parallel to the straight line passing through the midpoints of the segments *MA* and *MC.*

Point M is outside the plane of the regular pentagon *ABCOE.* Which sides or diagonals of the pentagon are parallel to the straight line passing through the centers of mass of the triangles *MAV* and *MAE7*

*M* and *N—*face centers *ABB \ A \* and *BCC \ B \* Cuba *ABCOA \ B \ C \ 0 \.* Which edges or diagonals of the faces of the cube are parallel to *MN?*

*AVSOEP.* closed polyline, not all of whose links are in the same plane. Sections connecting the midpoints of the links *Sun* and *AR, CO* and *EP* are equal and parallel. Are the links parallel *AB* and *OE ‘!*

*AWST).* square with a side of 6 cm.Dot *M* removed from each vertex of the square by 7 cm. Determine the distance from the middle of the segment *MA* to the midpoints of all sides of the square.

Perimeter of a regular hexagon *AVSOEP* is equal *R.* Point O, located outside the plane of the hexagon, is connected by a line segment to each of its vertices. From the center of mass of the triangle *OAV* drawn up to the intersection at points M), *Mch,* Mz, *M ^,* MB, MB with the plane of the hexagon straight lines, respectively parallel *OA, 0B, OS, 00, OE, OR.* Find the perimeter and area of a hexagon *M \ MhMgM ^ MbM ^.*

Three planes intersect in pairs. Prove that their intersections of the mowing line either intersect at one point or are parallel.

*ABCO.* 6 cm square, straight *AM* and *ST* are parallel. Such points are marked on them on one side of the square. *M* and *T,* what *MA: TS* 4: 3. At what distance from the vertices of the square is the point at which the line *MT* crosses the plane of the square?

Parallelism of a straight line and a plane

Planes b and a intersect. Prove that through each point of the plane b you can build a straight line that is either parallel to the plane *2* * MB* *2* * MC* *2* ^ (d 2 MO 2), where O is the center of the circle.

*MO.* perpendicular to the plane passing through

its point O. MA = 10 cm, MB *=* 16cm, ^ OAM = ^ 2OBM.

From the point M, which is outside the plane b, are drawn

perpendicular to this plane MO and oblique MA and MB.

Knowing that *AO =* 33 cm, *IN* = 8 cm, */. AMO* =. ^ *WMO,* find MO.

101 From point M, perpendicular are drawn to plane 6

MO and oblique *MA, MV, MS.* Projection *MV* and MC less projection *MA* at 33 and 48cm, *^ OAM: A. OWM* : ^ OCM = 1: 2: 3. Find MO.

Three perpendicular theorem

What figure do all points equidistant from

lines containing the sides of this triangle?

What shape do all points equidistant form

from three straight lines in plane b?

Right triangle legs *ABC 12* and 16 cm.Dot *M* removed from each of the lines *AB, AC, BC* 13 cm.Find its distance from the plane *ABC.*

Dot *M* removed from the top and sides of the right angle, respectively, 16, 12, 11 cm.Find its distance from the plane of the right angle.

On plane 6, an angle of 60 ° is given. Point M is removed from its vertex by 5. cm, and from the sides by 4 and 3 cm.Find the distance from point M to the plane of the named angle.

The bases of the rectangular trapezoid are 10 and 15 cm. Point M is 10 cm from each side of the trapezoid. Find the distance from point M to the plane of the trapezoid.

Points are marked on plane 6 *A and B, on.* plane a. points C and B so that *AB * 13 cm, CO = 14 cm, *AS* 8 cm, *BB* 17 cm, and straight *AS* perpendicular to planes 6 and Art. Find the distance between *AS* and *BB.*

If there is a point equidistant from all sides | parallelogram, then this parallelogram is a rhombus. Prove.

Perpendicular planes

What figure is formed by all the straight lines that pass through the vertex of a given angle (less than developed) and form ^ equal angles with its sides?

What is the shape of all points equidistant

from two given intersecting lines?

Rectangle *ABCV,* whose sides are 3 and 4 cm, bent diagonally *AS* so that the triangles *ABC* and *ABC* ended up in perpendicular planes. Determine the distance between points *B and B* after kink.

Planes “and p are perpendicular to plane 6. Prove that the line of intersection of planes a and p is perpendicular to plane 6.

Segment ends *AB* lie on two given mutually periendicular planes. Perpendiculars omitted *AA* 1 and *bv [* on the line intersection of the planes. Knowing that *AB* = | = 21 cm, *AA \ * 11 cm, *VVd * 16 cm, find *a \ b [.* I

Squares *ABCV* and *ABC \ B \* have areas of 32 cm 2. Knowing that the distance between *SV* and *C \ B \* is 8 cm, prove that the planes of the squares are mutually perpendicular.

Perpendicular planes intersect in a straight line *I.* Section *AB* has ends on these planes and forms angles of 30 ° and 45 ° with its projections. Find the angle between the directions *I* and *AB.*

*ABCO* square, plane *MAO* perpendicular to the plane of the square, *MM \\ JSC* On *AB* given a point *T.* How to build a straight line through this point, forming equal angles with AB and *Mt*

Perimeters of equilateral triangles *ABC* and *AVO* equal in 24 cm, the planes of the triangles are mutually perpendicular. Plot the common perpendicular to the medians *JSC* and OM of these triangles and find its length.

Square planes *ABCO* and an equilateral triangle *AVM* mutually perpendicular, *AB* *and.* Plot the common perpendicular line AC and median *MO* triangle and determine the length of this perpendicular.

Rectangular coordinates in space

The three vertices of the rhombus are located at points (8; 9; 10), (3, 3 “2), (8; 7, 1). Find the coordinates of the fourth vertex.

The three vertices of the parallelogram are at the points (3; 1-8), (4; 7; 1), (3; 5, 8). Find the coordinates of the fourth vertex.

The midpoints of the sides of the triangle are at the points (2; 5; 1), (1, 3; 4), (2 0; 4). Find the coordinates of the vertices of the triangle.

Vertex coordinates *A, C, E* regular hexagon *AVSOEP:* (-3; 7; 5), (7; 2; 1), (2; 3; 6). Find the coordinates of the remaining vertices and the center of the hexagon.

The sums of the applicate of the opposite vertices of the trapezoid are equal. Prove that the middle line of the trapezoid is parallel to the plane *hu* or is in this plane.

Do the points A (5;.1; 4), B (4;

Prove that line segments *AB* and *CO,* whose ends are at points *AND(“;*.one; 4), B (2; 8; 7), C (5; 0; 1), 0 (8; 6; 13), intersect and are divided in the ratio 1: 2.

On the ribs *AA ^, B \ C ^ CO* Cuba *ABCOA \ B \ C \ 0 \* find by point so that the sum of the squares of the distances between these points is minimal.

A straight line is drawn through the point M (1; 5; 3), which is parallel to the plane *hu* and intersects the segment with the ends A (4; 2; 1) and B (7; 11; 7). Determine the coordinates of the intersection point.

Find the point with the least sum of squared distances from points with coordinates (1; 2; 4), (4; 5; 1), (7; 2; 1).

Vectors in space

*ABCO.* rectangle, point *M* is outside its plane. Prove that *MA* *2* *MC* *2* *–* MV 2 MO 2

Dot *M* is outside the plane of the triangle *ABC,* whose center of mass is *T.* Dot *TO* divides the segment *MT* so that *MK * 3 *CT scan.* Prove that *AK VK SK MK* = 0.

If direction *AB* forms with directions *CO, CE, OE* equal angles, then a straight line *AB* perpendicular to the plane *COE.* Prove.

Is it true that if *M.* center of a regular polygon *A \ AchA ^. An,* then *MA \* *MAG* *MAz* *MAP* = O?

Find the point with the smallest possible sum of squared distances from all vertices of a given regular polygon.

Dot *M* away from the center of the cube *ABCOA \ B \ C \ 0 \* by 7 cm. Find the length of the vector *MA* *MV* *MC* four- *MO* *MA* ] *MV1 MS \* *M0 \. _*

By the condition of problem 135, find the length of the vector *MR \*

*МР2.МРз* M? 4 M? 5 *MRb,* Where? 1,? 2, pz, L, *P ^*

Re. the centers of the faces of the cube.

Through the center of mass *T* triangle *ABC* carried out. straight line, such points are marked on it *A \, B \, C \,* what *aa [* || II BB) || *SS \.* Prove that: a) la) BB1 *SS \* 0;

b) *TA ^* Tv, *TS \* 0.

Prove that the line passing through the points *AND* and B can be defined as a set of points *M,* satisfying the condition *AM = p AB,* where.oo 2. (y. Z) 2 = 25. Point *AND* has coordinates (0; 0; 5). Find such a point on the circle *IN,* so that the angle between *AB* and *hu* was the smallest possible.

*ABCV.* square, point *M* is outside its plane. Direct *VS t AS* form with plane *AVM* angles whose degree measures differ by and. Determine the values of these angles.

From point M, which is outside plane 6, inclined *MA* 23 cm and *MV* = 9 cm.Knowing that the angles between the inclined and plane b are related as 1: 3, determine the distance from the point *M* to plane b.

1 @ 6. From point *M,* removed from the plane 6 by 24 cm, two inclined ones are built, the lengths of which are related as 5: 8. The angles between the inclined and the plane are related as 1: 2. Find the lengths of the inclined.

From the point M ^ b, inclined *MA* and *MV,* whose projections onto the plane are 11 and 37 cm.Knowing that the angles between the inclined *“* plane are like 3: 1, find the distance from *M* until 6.

*MA* and *MV. n*inclined, forming with plane 5 containing points *AND* and *IN,* angles, one of which is 4 times larger than the other. Knowing that the projections of inclined onto this plane are 600 and 119 cm, find the distance from the point *M* to plane

From point *M* inclined to plane 8 *MA* and *MV* djnon 79 ‘and 25 em. The angles between the inclined and the plane will be measured as 1: 5. Find the distance from the point *M* to plane 6.

17 *r* _ 1 12 1. ^

ELEVENTH GRADE

Polyhedron

How many parts are the planes of all faces divided into: a) a triangular prism; b) a cube; c) triangular pyramid?

Draw a polyhedron with the total number of edges: a) 11;

Prove that no polytope has exactly 7 edges.

Draw a polyhedron other than a pyramid with as many vertices as there are faces.

5 points are given, no 4 of which lie in the same plane. Do the given points define a single polyhedron with vertices at these points?

Can there exist a polyhedron with an odd number of faces, and all of its faces are quadrangles?

Sometimes a prism is defined as a polyhedron, in which two faces are polygons lying in parallel planes, and all other faces are parallelograms. Give examples of the inaccuracy of this definition.

Draw a prism with as many vertices as diagonals.

Can an irregular prism have 4 planes of symmetry? If so, draw a prism that meets this condition.

The edge of the cube is 2 cm. The spider is in the center of the face of the cube. What is the shortest path over the surface of a cube a spider will have to make to get to the x vertex of a parallel face?

*AUSRERA \ B \ S1P \ ElR \.* prism. Prove that *AB \* BC) SD *A? 1 РЁ1 * YA.

The diagonal of the lateral face of the regular стистиstiangular prism forms an angle with the base plane that is 15 ° greater than the angle between the small diagonal of the prism and the base plane. Find these corners.

*AND* and *IN.* the midpoints of two non-adjacent lateral edges of a regular hexagonal prism. Find on the plane of the lower base of the prism such points that the straight lines *MA* and *MV* form equal angles with the plane of the lower base receptions.

Is it true that the area of the side face of a triangular prism is less than the sum of the areas of the remaining side faces?

The two side faces of the triangular prism are mutually perpendicular. Prove that the sum of the squares of the areas of these faces is equal to the square of the area of the third side face.

The three diagonals of the quadrangular tricks have a common point O. Prove that the fourth diagonal of the tricks also passes through the point O.

The base sides of a straight triangular prism are 5: 9: 10. The diagonals of the two smaller side faces are 26 and 30 cm. Find the area of the third side face.

The pedestal has the shape of a regular prism. Passing it, you can see 3 or 4 side faces. Determine the number of side faces of the pedestal.

The base of the prism is a right triangle *ABC,* two side faces *(ABB \ A \* and *АСС \ А \).* squares. Find ^ *B ^ ACx.*

Find the point with the least sum of squared distances from all vertices of a given regular triangular prism.

Prism surface area

Prove that the area of the side face of any prism is less than half the area of the side face of the prism.

The diagonal of the side face of a regular hexagonal prism is equal to the major diagonal of the base. Find the ratio of the lateral to full surface areas of the prism.

The distances of the lateral edges of the triangular prism from the parallel lateral faces are 12, 15, 20 cm; the smaller side face is square and perpendicular to the base plane. Find the surface area of the prism.

The area of the base and the area of the side faces of a straight triangular prism are respectively equal to 480, 312, 200, 128 cm 2. Find the height of the prism.

The base of the straight prism is a rhombus. Knowing that its height and diagonals are 40, 41, 50 cm, find the area of the lateral surface of the prism.

The base of a straight hexagonal prism is inscribed in a circle whose diameter is equal to the lateral edge of the prism. Three sides of the base, taken through one, are 5 cm long, the other sides are up to 3 cm.Find the surface area of the prism.

Height of a regular hexagonal prism *H.* The diagonals of two adjacent side faces drawn from one vertex are mutually perpendicular. Find the area of the lateral surface of the prism.

What is the largest lateral surface area that the correct *P-* a carbon prism with a side face diagonal and?

The base of a straight prism is a quadrangle inscribed in a circle with a radius of 25 cm.The areas of the side faces are as 7: 15: 20: 24, the length of the diagonal of the largest side face is 52 cm. Calculate the surface area of the prism.

Plane section of a prism

Prove that the section of a regular quadrangular prism by a plane passing through the ends of three edges outgoing from one vertex is an acute-angled triangle.

Two sections are drawn through the lateral edge of the triangular prism: one is perpendicular to the opposite lateral face, and the other is through its center. Knowing that the section planes divide the angle between the two side faces into three equal parts, find the values of the dihedral angles between the side faces of the prism.

Construct a section of the cube with a plane that is not parallel to any face of the cube, so that it has the shape of a square.

The edge of the cube is about. A section in the shape of a regular r-gon is constructed. For which *P* and how exactly can such sections be constructed? Calculate its area for each

Given a cube *AVSTA \ B \ C \ 1) \.* Construct a section of the cube with a plane passing through the midpoints of the edges *AB* and *Sun* parallel to the diagonal *B ^ \.*

The sides of the base of the triangular prism are 25, 39, 56 cm. The section passing through the center of the largest lateral edge and the lateral rib is square. Find the surface area of a prism.

In a regular quadrangular prism, the side of the base is 2 cm, the height is 4 cm. Find the cross-sectional area that passes through the midpoints of two adjacent sides of the base and the center of the prism (fig. 58).

Length of each edge of a regular hexagonal prism *ABC ^ EPA \ B \ C \ ^ \ E \ 1: ^ ‘\* 4cm. Find the area of the section that goes through the vertices *AND* and C parallel to the diagonal of the prism *BE ^.*

In a regular quadrangular prism *ABCBA ^ B \ C \ B \* side face and section *AB \ C* equal areas. Find the angle between the plane of the named section and the side edge of the prism.

The plane intersects the lateral edges of a straight triangular prism *ABCA \ B \ C \* so that the section turned out to be an equilateral triangle *KLM* perimeter 36 cm. It is known that *AK =* 16 cm, *Bb =* 11 cm, *CM* = 5 cm. Find the angle between the median *Kv* section and base plane (Fig. 59).

In a regular quadrangular prism, two parallel sections are constructed: one through the midpoints of two adjacent sides of the base and the center of the prism, the other through the diagonal of the base (Fig. 60). Find the ratio of the cross-sectional areas.

Parallelepiped

The section of a prism by a plane intersecting all lateral edges is a parallelogram. Prove that this prism is a parallelepiped.

Side edge of a rectangular parallelepiped *I,* its diagonal is half the perimeter of the base. Determine the area of the base of the box.

Prove that in a rectangular parallelepiped the square of the cross-sectional area with vertices at the ends of the edges emanating from one vertex is 8 times less than the sum of the squares of the areas of all faces of the parallelepiped.

Prove that the distance between crossing diagonals of two adjacent faces of a cube is three times less than the diagonal of a cube.

Prove that the sum of the squares of the diagonals of a parallelepiped is equal to the sum of the squares of its edges.

Distances from the center of the box to its vertices are 18, 15, 11, 10 cm.Knowing that the lengths of three edges (in centimeters) are expressed in sequential integers, determine the perimeters of the sides of the box.

The side edge of the parallelepiped is 10 cm, the perimeter of the base is 56 cm. The distances from the vertices of one base to the center of the other base are 18, 17, 10, 9 cm. Find the sides of the base.

Diagonals of a parallelepiped *ABCOA \ B \ C \ B \* intersect at the point *ABOUT.* Perimeters of triangles *ОАА \, ОАВ* and *JSC* equal 36, 37, 29 cm, AL, 17 cm, *AB =* 11 cm, *AO =* 6 cm. Find the diagonals of the parallelepiped.

The side edge of the parallelepiped is 3 cm, the sides of the base are 10 and 11 cm.Knowing that the lengths of the diagonals (in centimeters) are expressed in consecutive even numbers, find the areas of the diagonal sections.

The lengths of the edges of the parallelepiped are 9, 13, 14 cm, the lengths of its diagonals (in centimeters) are expressed in consecutive even numbers. Find the distances from the center of the box to the vertices.

The surface area of the rectangular parallelepiped is 192 cm 2. If each dimension of it were 1 cm larger, the surface area would be 274 cm 2. Determine the length of the parallelepiped’s diagonal.

Prove that a section of a parallelepiped by a plane cannot be a regular pentagon.

What is the largest surface area that a rectangular parallelepiped with the length of the diagonal and?

What is the largest surface area that a parallelepiped can have, in which the sum of the lengths of all edges is 48 cm?

Can the midpoints of all heights of a triangular pyramid be in the same plane??

The sum of plane angles at all vertices of a pentagonal prism is equal to the sum of plane angles at all vertices of the pyramid. Determine the number of edges of this pyramid.

The flat angles at each top of the pyramid are equal to each other. Determine the shape of the base of the pyramid.

Whatever the triangular pyramid, you can build a triangle, the sides of which are equal to the sum of the crossing edges of this pyramid. Prove.

Prove that the line segments connecting the midpoints of the crossing edges of the triangular pyramid intersect at one point.

Prove that the sum of the squares of the line segments that connect the midpoints of the crossing edges of a triangular pyramid is 4 times less than the sum of the squares of the edges of this pyramid.

Can all the faces of the pyramid turn out to be right-angled triangles?

The flat angles at the top of the pyramid are straight. Prove that the sum of the squares of the areas of the side faces is equal to the square of the area of the base of the pyramid.

The base of the pyramid is a parallelogram, the sides of which are 16 and 22 cm.The distance from the top of the pyramid to the center of the base is 4 cm.Knowing that the lengths of the side ribs (in centimeters) are expressed in consecutive odd numbers, find the lengths of the side ribs of the pyramid.

The two side edges of the pyramid are 13 and 14 cm, the angle between them is 60 °, and between their projections is 120 °. Find the height of the pyramid.

The base of the pyramid is a parallelogram, the perimeter of which is 48 cm. The center of the base is 7.5 cm from the top of the pyramid, the side edges of the pyramid are 9, 11, 12, 13 cm. Find the sides of the base.

Can a scan of the full surface of the pyramid turn out to be: a) an equilateral triangle; b) a square;

c) a regular pentagon; d) a regular hexagon; e) trapezoid?

6T. Prove that the centers of all faces of a regular prism are the vertices of two equal regular pyramids with a common base.

6c. Prove that only for *P* 3 full surface sweep

an n-sided pyramid may turn out to be a convex polygon.

© 9. If the flat angles at the top of the pyramid are straight, then the height of the pyramid passes through the intersection of the base heights. Prove.

T®. The base of the pyramid is a square. The dihedral angles at the base of the pyramid are in the ratio 1: 2: 5: 2. Find the values of these angles.

Lateral rib of a regular triangular pyramid *MAVS* has length *I* and forms an angle of 75 ° with the side of the base that it crosses. The spider began to crawl from the top *AND* and having visited all the side faces of the pyramid, he returned to the same point (Fig. 61). Determine the smallest possible spider path.

Base side of a regular hexagonal pyramid *MAVSVER* equals *and,* the angle between the side edge and the side of the base that it crosses is 80 °. The spider began to crawl along the surface of the pyramid from a point *AND* and, having visited all the side faces, returned to the point *AND.* Determine the smallest possible spider path.

From each vertex of the base of a regular quadrangular pyramid, the base area of which is *ABOUT,* omitted perpendiculars on the plane of faces that do not contain these vertices. The intersection points of these perpendiculars are *K, b, M, N* (fig. 62). Prove that these points lie in the same plane and find the area of a quadrilateral *K ^ MN.*

If the lateral edges of a triangular pyramid are mutually perpendicular in pairs and have lengths *a, b, c,* then the height of the pyramid *H* associated with them by the ratio: *H* *2* *from*

*2* *.* Prove.

If the sums of the squares of the crossing edges of a triangular pyramid are equal, then the heights of the pyramid intersect at one point Γ. Prove

Surface area of pyramids

The side edges of the triangular pyramid are mutually perpendicular, their lengths are 2, 4, 16 cm.Find the surface area of the pyramid.

The area of the base of the triangular pyramid is 56 cm 2. The lateral edges are mutually perpendicular, their lengths are arithmetic progression with a difference of 4 cm. Find the area of the lateral surface of the pyramid.

What is the largest surface area a triangular pyramid can have, in which 5 edges have length a?

Dihedral angle between adjacent lateral faces of a regular quadrangular pyramid 120 °, base area O. Determine the area of the lateral surface of the pyramid.

In a regular hexagonal pyramid, the area of each diagonal section is O. Find the area of the side and the area of the total surface of the pyramid.

A regular pyramid and a regular prism have a common base and height. Can the area of the lateral surface of the prism be less than the area of the lateral surface of the pyramid? If yes, ‘then under what condition?

Can the area of one side face of the pyramid be equal to the sum of the areas of the other side faces? Can it exceed the named sum of areas? Support your considerations with examples.

The lateral surface area of a regular quadrangular pyramid is equal to the sum of the areas of the base and the diagonal sections. Find the value of the plane angle at the top of the pyramid.

From the center of the base О of a regular quadrangular pyramid, the surface area of which is *ABOUT,* parallel to the side edges of the pyramid, straight lines *OA \, OV \, OS \, OV \* (fig. 63). Find the surface area of the pyramid *OA1B \ C \ B \.*

Section of the pyramid

The plane angle at the top of a regular pyramid is straight. How to construct a section of a pyramid with a plane passing through the top of the pyramid so that it is an equilateral triangle?

The side of the base of a regular triangular pyramid is 20 cm, the side edge is 30 cm.Construct a section in the shape of a square and determine its area.

The area of the small axial section of a regular quadrangular pyramid O. Find the area of the section that is perpendicular to the side of the base and divides this side in a ratio of 1: 5.

In a regular hexagonal pyramid, the side of the base is 10 cm, and the side edge is 13 cm.Find the area of the section passing through the center of the base parallel to the side face.

Base side of a regular quadrangular pyramid *MAWSO* equal to a, lateral edge *I.* Draw a section through the midpoints of the sides of the base *AB* and *Sun* parallel to the rib *MV* and determine the cross-sectional area.

The base side of a regular quadrangular pyramid is 12 cm, and the side edge is 11 cm.Find the cross-sectional area passing through the base side perpendicular to the opposite side face.

The perimeter of the base of a regular triangular pyramid is 45 cm, the lateral edge is 14 cm.Find the area of the section that passes through the middle of the base median perpendicular to this median.

Through the side of the base of the regular quadrangular pyramid and the middle line of the parallel side face of the-

a cross-section has been made. Prove that its area is larger. areas

Through the side of the base of the regular hexagonal pyramid and the middle line of the parallel side face of the-

the plane is drawn. Prove that the cross-sectional area is larger.

Base of the pyramid *MAVSV.* rhombus with diagonals *AS* = 24 cm, *IN* 21cm. Side rib *MA* 18 cm perpendicular to the base plane. Find the cross-sectional area that passes through the vertex *AND* and the middle of the rib *MC* parallel to the diagonal *IN* bases (fig. 64).

Parallel sections of the pyramid

Two sections of the pyramid were constructed with planes perpendicular to the side edge. Are the areas of these sections related to the squares of their distances from the top of the pyramid??

The area of the base of the pyramid is 128 cm 2. The areas of two sections parallel to the base are 18 and 50 cm 2. The distance between the planes of the sections is 12 cm. Find the height of the pyramid.

The lateral edge and the height of the regular quadrangular pyramid are 35 and 28 cm. A cube is inscribed into the pyramid so that its 4 vertices lie on the base of the pyramid, and 4 on the apothems of the pyramid. Find the edge of the cube.

The base of the pyramid is a right-angled triangle with 3 and 4 cm legs. The height of the pyramid *H * 24 cm is inside the pyramid. A cube is inscribed into the pyramid so that 4 of its tops lie on the base of the pyramid, and 4 on the side faces, and the side faces of the cube are parallel to the legs of the base (Fig. 65). Find the edge of the cube.

Truncated pyramid

Prove that the diagonals of a regular rectangular truncated pyramid meet at one point.

The areas of the bases of the truncated pyramid are 75 and 147 cm 2. Find the cross-sectional area passing through the midpoints of all lateral ribs.

The diagonal of a regular rectangular truncated pyramid is 15 cm long and divides the segment connecting the centers of the bases into 4 and 5 cm pieces.Find the areas of the bases of the truncated pyramid.

Section *00 \* = 27 cm, connecting the centers of the bases of a regular quadrangular truncated pyramid, divided its diagonal into parts of 20 and 25 cm.Find the areas of the bases.

The side of the smaller base, the side edge and the side of the larger base of a regular rectangular truncated pyramid make an arithmetic progression with a difference of 4 cm. The height of the truncated pyramid is 7 cm. Find the areas of the bases.

In a regular hexagonal truncated pyramid, the segment connecting the middle of the small diagonal of the larger base to the center of the other base is parallel to one of the side edges. How are the base areas of a truncated pyramid??

In a regular triangular truncated pyramid, the sides of the bases are 2 and 5 inches, the height is 1 inch. Find the cross-sectional area passing through the side of the smaller base parallel to the side rib.

The sides of the bases of a regular triangular truncated pyramid are 1: 3. The perimeter of the side face is

perimeter of one of the bases. Find the angle between the side edge and the plane of the base.

The center of each base of a regular triangular truncated pyramid is connected to the vertices of another base (Fig. 66). Find the length of the mowing line that connects in pairs the intersection points of the constructed segments if the perimeters of the bases of the truncated pyramid are equal *R* and *R\.*

Truncated pyramid surface area

The sides of the base are more regular than the hexagonal truncated pyramid 5 and 11 cm.The distance between the parallel sides of the bases that do not lie in one face is 19 cm.Find the surface area of the truncated pyramid.

The section passing through the midpoints of all lateral edges of the regular pyramid divided it into parts, the areas of the full surfaces of which are in the ratio 3: 11. Determine the dihedral angle at the base of the pyramid.

The perimeters of the bases of a regular triangular truncated pyramid are 18 and 36 cm. The distance from the apex of the smaller base to the opposite side of the other base is 7 cm. Find the area of the lateral surface of the truncated pyramid.

Perimeters of the bases of a regular hexagonal truncated pyramid *AVSVERA \ B1C \ B \ E \ R \* 28 and 124 cm. Distance from the top *AND \* smaller base to straight *CE* equals 17 cm.Find the area of the lateral surface of the truncated pyramid.

The bases of the truncated pyramid are rhombs with a side ratio of 3: 4 and side lengths of 15 and 25 cm. One of the lateral ribs is perpendicular to the base plane and is equal to the smaller diagonal of the smaller base. Find the surface area of the truncated pyramid.

Regular polyhedra

Prove that a tetrahedron with vertices at the centers of mass of the faces of a regular tetrahedron is regular. How are the surface areas of these tetrahedra related??

In what ratio are divided when crossing the height of a regular tetrahedron?

For which *P* it is possible to construct a cross-section of an octahedron by a plane, which is a regular ge-angle?

Prove that the degree measures of the dihedral angle of a regular tetrahedron and the angle between adjacent faces of the octahedron add up to 180 °.

Point O. mid-height *MO* regular tetrahedron *MAVS.* Prove that the rays *OA, 0V, OS* pairwise mutually perpendicular.

How many centers of symmetry have two parallel planes? What shape do all these centers form??

Build a figure symmetrical to this triangular pyramid about its center of mass: a) base; b) this side face.

Build a figure symmetrical to the dyna regular hectare pyramid (n 4; 6; 3) relative to the middle: the height of the pyramid.

*ABCVA \ B \ C \ B \.* parallelepiped, point *M* 6 *al.* Draw a line *МN,* whose middle is on the plane *SS \ A,* and point *N* lies on the edge *SV.*

Draw a line segment with ends on edges *AB* and *MC* and the middle at a height *MO* correct pyramid *MAVS.*

Prove that any quadrangular pyramid can be intersected by a plane so that the section has a center of symmetry.

Write an equation for a plane that is symmetrical to the plane *x y. \. z.* 3 = 0 relative to point *M* (2; 2; 2).

Given a square *ABCB with* peaks *AND* (4; 0; 0) and *IN* (8; 3; 0), the plane of which is parallel to the *Og.* Find the coordinates of the vertices of the square that is symmetrical to the given one about the point (2; 2; 2).

*MAVSV.* correct pyramid. Build a figure symmetrical about the plane of the base: a) the middle mowing line of the side face (two cases); b) a segment connecting the centers of mass of the faces *MAV* and *MVS;* c) faces *MAO.*

*ABCA \ B \ C \.* correct reception. Draw a shape that is symmetrical about a plane *ABB \*: a) segment *B ^ ‘,* b) a given segment with ends on *The EU* and *A \ C \.*

13B. All the edges of the pyramid *MAVSV* are equal. Find on the plane of its base a point equidistant from points P and Y lying on *MA* and *MC.*

Points *IN* and *E* are on the side faces of a regular pyramid *MAVS.* Find on the plane *ABC* point with the smallest possible sum of distances *from B and E.*

Points *IN* and *E* are at the height of the triangular pyramid *MAVS.* Construct all points equidistant from points on the surface of the pyramid *IN* and *E.*

Points *In ta E* are on the side of the base of a regular pyramid *MAVS.* Find on the surface of the pyramid all points equidistant from *IN* and *E.*

Points *IN* and *E* are on the verge *MAV* and *MVS* the correct pyramid ML.V.S. Construct an isosceles triangle with apex at *MV,* base ends on *AB* and *Sun,* so that the sides contain *In u. E.*

Points *E* and *R* are on the verge *MAV* and *IES* regular quadrangular pyramid *MAWSO.* Build an isosceles trapezoid with one base on the base of the pyramid and the ends of the other on the edges *MV* and *MC,* and the sides contain points *E* and *R.*

*AVSVERA \ B \ C \ B \ E \ R \.* correct prism. Construct on its surface all points belonging to the plane of symmetry of the planes: *a) AA \ B and CC \ P ‘,* b) *AA \ B* and *AA \ E;* in) *AA \ B* and *AA \ B;* d) *AA ^ B* and *BB \ C;* e) *AA ^ C* and *BB ^ P; f) AA ^ B* and *BB \ E;*

g) *AA* ,With and *BB \ R.*

Equality of spatial figures

Are two triangular prisms equal if the three sides of the base and the side edge of one are equal to the three sides of the base and the side edge of the other? If not, what additional condition is needed to assert that the prisms are equal?

Two pyramids have equal heights, their common base is a square *ABCO.* Prove that these pyramids are equal if their vertices are orthogonally projected: a) into points *AND* and C; b) the middle of the two sides of the square.

*avsva \ v [s \ v \* *–* cub. Prove that the pyramids *ABCV \* and *1) B \ C \ B \* are equal.

Formulate several signs of equality of correct prisms. Justify these signs.

Formulate several signs of equality of regular pyramids. Justify these signs.

Prove that two triangular prisms are equal if their side faces are respectively equal.

Are two straight triangular prisms equal if all the diagonals of their side faces are respectively equal?

*MAVSVER.* correct pyramid. Prove the equality of the pyramids: a) *MAVS* and *MVER;* b) *MFE* and *MARV.*

*ABC ^ EPA ^ B ^ C ^^^ E ^ P ^.* correct prism. Are pyramids equal: a) *С ^ ВСВ* and *EE \ B \ P ^,* b) *A ^ ABP* and *C \ CBE;*

in) *BAA ^ B* and *A ^ SS ^ HF*

What shape do all the points distant from this line form? *I on. and* equidistant from these points *AND* and in?

Construct an image of a regular octagon and a regular dodecagon inscribed in a circle.

Draw a right-angled triangle inscribed in a circle with a leg ratio of 2: 3.

Draw two equal chords of a circle intersecting at a given point *M* at right angles.

Draw two equal chords of a circle intersecting at a given point *M* at an angle of 60 °.

Draw a tangent line to a given ellipse at a given point of this ellipse.

Construct images of a rhombus circumscribed about a circle with an angle of 45 ° and an isosceles trapezoid with an angle of 45 ° with a larger base.

The vertices of the rectangle lie on the circumferences of the bases of the cylinder, whose radius is 13 cm, and the generatrix is 32 cm.Knowing that the sides of the rectangle are 1: 4, find its area.

The diagonal of the axial section of the cylinder is equal to the sum of its radius and height. Find the aspect ratio of the axial section of the cylinder.

The diameter of the winch drum is 530 mm, its length is 727 mm. During operation, 225 m of 17 mm diameter cable is wound onto the drum. How many layers is the cable wound?

About this cylinder, describe a regular quadrangular pyramid, the height of which is twice the height of the cylinder.

The height and base of the isosceles triangle are 8 and 6 cm. The cylinder touches all sides of the triangle, its generatrices are inclined to the plane of the triangle at angles of 30 °. Find the radius of the cylinder.

Find the radius of an equilateral cylinder whose axis lies on the diagonal of a cube with an edge *and,* and each of the base circles touches three cube faces that have a common vertex.

Into an equilateral cone, the generatrix of which *I,* a regular hexagonal prism is inscribed, the side face of which is a square. Find the areas of the diagonal sections of the prism.

Diagonal octahedron with edge *and* is the height of the cone, on the surface of which there are 4 edges of the octahedron (Fig. 67). Find the area of the axial section of the cone.

The radius of the base of the cone is 9 cm, the height is 7 cm. What is the largest area of the section of the cone with a plane passing through the apex of the cone?

The largest possible cross-sectional area of the cone by a plane passing through the apex of the cone is twice the area of the axial cross-section. Find the angle between the generatrix and the plane of the base of the cone.

A regular triangular prism is inscribed into the cone, all edges of which are equal *and.* The four vertices of the prism lie on

the circumference of the base, and two on the lateral surface of the cone (Fig. 68). Find the height of the cone.

Cube edge *ABCBA \ B \ C \ B \* equally *and.* Diagonal *AC \* contains the height of an equilateral cone with apex *AND.* The circle of the base of the cone touches the three faces of the cube with a common point C1. Find the generator of the cone.

The base of the cone is on the edge *ABCV* Cuba *ABCVA \ B \ C \ B \, y* whose rib *and.* The vertex of the cone is at the center of the face *A \ B \ C \ B \.* Find the radius of the base of the cone, knowing that the lateral surface touches the straight line that passes through: a) *IN\* and the middle *Sun;* b) B and middle *ВС \;* c) the middle *Sun* and *BЁ1* (fig. 69).

Frustum

What is the shape of the midpoints of the diagonals of all axial sections of the truncated cone??

What shape is formed by the midpoints of all segments, each of which has the ends on the circumferences of the bases of the truncated cone?

The radii of the bases of the truncated cone are 25 and 16 cm. A circle can be inscribed in the axial section of this truncated cone. Define *her* radius.

The diagonal of the axial section of the truncated cone is divided

the axis of the truncated cone into parts in 13. and 26. see. Knowing,

that the generatrix of the truncated cone is 26 cm, find the radii of the bases.

Two cones, whose base radii are 10 and 15 cm, have a common height; their base planes do not coincide. Find the circumference along which the surfaces of these cones intersect.

Sphere and ball

What figure is formed by the bases of the perpendiculars dropped from a given point *AND* to all planes passing through this point B?

From point *M* three mutually perpendicular tangents can be drawn to this ball. What figure do all such points M?

Which figure is formed by the points that are located at a distance from a given sphere of radius b?

What shape is formed by the centers of all spheres of radius *IN,* touching: a) a given plane 6 ^ b) two given planes?

Given plane b and point *M* outside of it. What shape is formed by the centers of spheres of radius *IN,* which pass through point M and touch plane b?

Prove that the tangents from a given point to a given sphere have equal lengths.

Plane 6 touches the ball at a point *AND.* On the continuation of the diameter *AB* = *and* a point C is taken such that *Sun* *B,* it contains a point light source. Find the area of the ball’s shadow on plane 6.

Diameters *AB, SV, EP* the spheres are mutually perpendicular. Each one is divided into *P* equal parts, planes perpendicular to this diameter pass through the division points. How many parts did these planes divide the sphere into if: a) *P* four; b) *P* 6; in) *n.- =-* five; d) *P* 8?

In a ball with a radius of 18 cm, two mutually perpendicular sections are drawn, the radii of which will slope like 2: 3. Knowing that the total chord of these sections is 2 cm, find the cross-sectional areas.

Two mutually perpendicular sections are built in the ball, the areas of which are 185 and 320 cm 2. Determine the radius of the ball if the total chord of these sections is 16 cm long.

Draw a triangular pyramid inscribed in a sphere, the side edges of which are mutually perpendicular.

Into a sphere of radius *H* a regular hexagonal prism is inscribed. The radius of the sphere, drawn to the apex of the prism, forms an angle of 30 ° with the plane of the side face. Find the area of the lateral surface of the prism.

Plane angle at the apex of a regular triangular pyramid. straight, side of the base *and.* Find the radius of the circumscribed sphere.

Prove that the radius of the sphere described by the pyramids-

dy whose height *H,* and each side rib *I,* is equal. *t*

Establish at what ratio between *I* and *H* the center of the described sphere is inside the pyramid.

At the triangular pyramid *MAVS: MA VS* = 16 cm, *MV* *AS* = s 19 cm, *MC* *AB* 21 cm. Determine the radius of the described sphere.

The radii of the circles circumscribed near the base and near the lateral face of the regular triangular pyramid are 8 and 7 cm. Find the radius of the circumscribed sphere.

A ball can be inscribed in a rectangular prism. Is it true that the sums of the areas of its opposite side faces are equal?

Crossed edges of a tetrahedron are equal in pairs. Prove that the centers of the circumscribed and inscribed spheres coincide.

All edges of a quadrangular pyramid are equal *and.* Find the radius of the sphere that touches all the edges of the pyramid.

Each edge of the tetrahedron has a length *and.* Find the radius of the sphere that touches all the edges of the tetrahedron.

A ball is inscribed into the cone, which has a base radius of 9 cm and a generatrix of 15 cm. Find the length of the mowing line that touches the surface.

Sphere and its equation

The radii of the two balls are 17 and 25 cm. The length of the mowing line along which the surfaces of the balls intersect is 30 liters. Determine the distance between the centers of the balls.

There is a fragment of the ball. On the basis of what constructions and measurements you could determine its radius?

Set the relative position of the spheres x 2 *at* *2* r 2 = 4 and *x* *2* at 2 *r* *2* *–* 24g. 12u 16g. 168 = 0.

Establish the relative position of the sphere *x* *2* y 2 4- 2 2 16 and plane *2x. 2y* 2. 12 0.

Write the equation of the sphere that passes through the points (2; 3; 4) and (3;.1; 5) and touches the plane *hu.*

Volume of a rectangular parallelepiped

How, having cut into two parts, fold a cube from a rectangular parallelepiped, the dimensions of which are: a) 4, 6, 9 cm;

b) 9, 12, 16 cm; c) 16, 20, 25 cm?

What is the smallest number of parts that a cube can be cut into, so that a prism can be folded from these parts, the base of which is: a) a rectangular trapezoid; b) isosceles trapezoid?

Find the volume of a rectangular parallelepiped whose center-to-edge distances are 13, 20, 21 cm.

On the ribs *AA \* and *BB \* rectangular parallelepiped *ABCVA \ B \ C \ B \* given points *M* and *N.* Construct a plane that passes through these points and divides the parallelepiped into equal parts.

Solve Problem 199 for the case when points are given on adjacent side faces.

The lengths of the edges of the four cubes (in centimeters) are expressed in sequential whole numbers. The volume of one cube is equal to the sum of the volumes of the others. Determine the lengths of the edges of these cubes.

Prove that of all rectangular parallelepipeds with a given diagonal length, the cube has the largest volume, using the theorem: “The product of three positive numbers, the sum of which is constant, has the largest value when these numbers are equal.”.

Find the volume of a rectangular parallelepiped with the perimeters of three faces 36, 40, 48 cm.

Find the volume of a rectangular parallelepiped with a diagonal length of 81 cm, and the dimensions are as 7: 14: 22.

Volume of a straight parallelepiped

In a right parallelepiped *ABC ^ A \ B \ C ^^ \* diagonals *AC \* and *B ^ \* mutually perpendicular and equal to 6 and 8 dm. Knowing that *Sun* 3 dm, find the volume of the parallelepiped.

The dihedral angle between the lateral faces of a straight parallelepiped is 60 °, the area of diagonal sections is 56 and 72 cm 2. The length of the lateral edge is 4 cm. Find the volume of the parallelepiped.

Distances from the center of the straight line, parallelepiped to the base and side faces 9, 8, 6 cm. Base perimeter P = 70 cm. Determine the volume of the parallelepiped.

The surface area of a straight parallelepiped is 176 cm 2. The distance from the center of the parallelepiped to its faces is 1, 2, 3 cm. Find the volume of the parallelepiped.

The volume of the oblique parallelepiped

The base of the parallelepiped is a rectangle with sides about and *B.* Side rib equals *I* and forms angles of 45 ° and 60 ° with the sides of the base. Find the volume of a parallelepiped.

Each face of the parallelepiped is a rhombus with diagonals of 6 and 8 inches. Flat corners at one vertex are sharp. Find the volume of a parallelepiped.

Prism volume

The area of the base of a regular quadrangular prism O. The lengths of the diagonals of two faces are in the ratio 1: 3.

Find the volume of the prism.

A reinforced concrete silo tower made of standard slabs has the shape of a regular prism, in which the distance from a straight line passing through the centers of the bases to the walls is 3.65 m.Knowing that the volume of the walls is 6.45% of the usable volume,

determine the thickness of the walls.

Two regular prisms of the same name are given. On one side of the base *and,* lateral rib *B,* on the other side of the base *B,* lateral rib *a (a* *B).* Which of the prisms has more volume?

Two eponymous regular ha-coal prisms of the same size. Which one has more lateral surface area

The cross-section of the channel is a trapezoid (without the upper base), the bottom and walls of the channel with a length of about. At what value of the angle between the bottom and walls of the channel, its throughput will be the greatest?

The lateral face area of a regular hexagonal prism ‘O. The plane passes through the side edge and divides the prism into parts, the volumes of which are 1: 3. Find the cross-sectional area.

Height of a regular hexagonal prism *H.* The angle between two equal diagonals of a prism drawn from one vertex is 30 °. Find the volume of the prism.

The base of the straight prism is a trapezoid, the perimeter of which is 58 cm. The areas of parallel lateral faces are 96 and 264 cm 2. and the areas of the other two lateral faces are 156 and 180 cm 2. Find the volume of the prism.

The base of the straight prism is a trapezoid, the area of which is 306 cm 2. The areas of parallel lateral faces are 40 and 300 cm 2. and the areas of the other lateral faces are 75 and 205 cm 2. Find the volume of the prism.

The base of the straight prism is a quadrangle inscribed in a circle with a radius of 65 cm. The areas of the lateral faces are as 63: 52; 39: 16. The diagonal of the smallest side face is 40 cm. Find the volume of the prism.

A hexagonal prism is inscribed into a cylinder 12 cm high, in which three sides, taken by one worm, have lengths of 3 cm, the other sides of the base. 5 cm each.Find the volume of the prism,

An octagonal prism is inscribed in a cylinder 8 cm high, in which the lengths of the four sides of the base, taken through one, are 2 cm each, and the other sides of the base are 3 cm each. Find the volume of the prism.

A regular triangular prism is inscribed in a sphere of radius L. The radius of the sphere drawn to the vertex of the prism is inclined to the plane of the side face at an angle k. Find the volume of the prism.

Pyramid volume

### Geometry 2: Angles and Polygons (shapes)

The sides of the base of the triangular pyramid are 15, 16, 17 cm. Each side edge is inclined to the base plane at an angle of 45 °. Find the volume of the pyramid.

226, The length of each side edge of the pyramid is 65 cm. Its base is a trapezoid with side lengths of 14, 30, 50, 30 cm. Find the volume of the pyramid.

The length “of each side edge of the pyramid is 35 cm, the sides of the base are 20, 34, 60, 66 cm. Find the bypass of the pyramid.

The height of the correct vdetiuronal pyramid I, the distance from the middle of the height de lateral rib *at* 4 times smaller than the side of the base. Find the volume of the pyramid.

The length of the heel of the ribs of a triangular pyramid is no more than 2 cm. Prove that the volume of the pyramid is no more than 1 cm 3.

Prove that the volume of a triangular pyramid is less

The sides of the base of the truncated ‘triangular prism are 28, 45, 53 cm, and the side edges are perpendicular to the base and equal to 13, 14, 15 cm. Find the volume of the truncated prism (fig. 70).

If a plane that is not parallel to the plane of the base of the prism intersects all the lateral edges of the prism, then the resulting parts of the technique will be called truncated prisms.

Prove that the volume of a truncated triangular prism is equal to the product of the area of the perpendicular section by the arithmetic mean of the lengths of the lateral edges.

The sides of the base of the straight parallelepiped are 6 and 8 cm, the angle between them is 30 °. The plane cuts off segments of 8, 10, 11 cm on three lateral edges.Find the volume of that part of the prism that is enclosed between the base and the section plane.

Base of a straight prism *–* trapezoid, whose sides *AB* CO 13 cm, *Sun* = 18 cm, *AT * 28 cm.The plane passes through point C and cuts off at the edges *BB \* and *BB \* segments of 9 cm.Find the volume of the part of the prism between the base and the cut.

In a parallelepiped *ABCVA \ B \ C \ 0 \* dot *K.* mid rib *AA \,* point M. midpoint *SS \, BB \* = *a, KV \* *B, MV \ s,* moreover *BB \, KV \* and MB1 are mutually perpendicular in pairs. Find the volume of a parallelepiped.

Flattened pyramid surface. square with side *and.* Find the volume of the pyramid.

The lengths of the sides of the base of the triangular pyramid are 32, 34, 34 cm. The perimeters of two equal side faces are 150 cm each, the third is 162 cm. Find the volume of the pyramid.

Given tetrahedrons *MAVS* and *M \ A \ B \ C \, y* which trihedral angles with vertices M and M1 are equal. Prove that the volumes of these tetrahedra are related as products of the lengths of the edges of equal trihedral angles.

A plane is drawn through the side of the base and the middle line of the opposite side face of the regular quadrangular pyramid. Find the ratio of the volumes of the parts into which the plane divided the pyramid.

A plane is drawn through the side of the base and the middle of the height of a regular quadrangular pyramid. Find the ratio of the volumes of the parts into which the pyramid is divided.

The unfolded pyramid is an isosceles triangle with a base 18 cm and a height drawn to the base 12 cm.Find the volume of the pyramid.

Prove “that the volume of the regular pyramid is less

-the cube has the length of its lateral rib.

Each side edge of the pyramid *MAVSV* equally *I.* It is known that ^ *AMB = /. Navy* ^. *AMC * 90 °, *^ AMO = ^ CMB.* Find the volume of the pyramid.

The base of the pyramid is a trapezoid (or triangle) with a middle line *AB,* top of the pyramid M, *ABOUT*. the middle of the side parallel to the middle line. Prove that the volume

the pyramid is equal to the product of the sectional area *MAV* on z

distance from point O to plane *MAV* (fig. 71).

The bases of a polyhedron lie in parallel planes, all other faces are triangles or trapezoids, all of whose vertices lie on the bases. Prove,

that the volume of the polyhedron *V =.* ((?one *Og * four *at* *” *

Prove that the volume of the spherical segment is equal to yay 2 (lz-) where u is the radius of the ball, and *H.* segment height.

A ball is inscribed in the cone, whose base radius is 6 cm, and the generatrix is 10 cm. A plane is drawn through the line of contact of these bodies. Find the ratio of the volumes of the parts into which this plane divides the ball.

A ball with a radius of 9 cm floats in water, the height of the part protruding from the water is 6 cm.Find the density of the material from which the ball is made.

A hemispherical vessel filled with water. What part of the water will pour out if the vessel is tilted by: a) 30 °; b) 45 °?

The height of an equilateral cone is *H* and is the diameter of the ball. Find the volume of the part of the ball that lies outside the cone.

Cylinder surface area

All edges of a regular triangular prism are equal *and.* Side edges are the axes of cylindrical surfaces

radius. ^. Calculate the surface area of a body bounded by named cylindrical surfaces and prism bases.

A quadrangular prism is inscribed in the cylinder, with the perimeters of the side faces 30, 45, 56, 64 cm.Knowing that one of the diagonal sections of the prism contains the axis of the cylinder, find the total surface area of the cylinder.

Cube edge length *and.* The axis of an equilateral cylinder lies on the diagonal of the cube. Each circle at the base of the cylinder touches three faces of the cube. Find the volume and surface area of a cylinder.

Cone surface area

The cylinder and cone have a common base and a common height. The areas of their total surfaces are 7: 4. Find the angle between the generatrix and the plane of the base of the cone.

A quadrangular pyramid is inscribed into the cone, with the perimeters of the side faces 78, 94, 104, 112 cm. One of the diagonal sections of the pyramid contains the height of the cone. Find the surface area of the cone.

Square *ABCV* an area of 120 cm 2. bent, placed on the surface of the cone. In this case, the diagonal *AS* coincided with the generator, and the diagonal *IN* appeared on the lateral surface of the cone and its ends coincided (Fig. 74). Determine the volume and surface area of the cone.

Hemisphere radius *H.* On the basis of the hemisphere, a cone is built, each generatrix of which is divided by the surface of the hemisphere in a ratio of 1: 2, counting from the top. Find the surface area of this cone.

A cone of the largest possible volume is inscribed in a sphere of radius L. Determine the surface area of this cone.

The radius of the base of the cone is L. The sphere touches the base of the cone and divides each generatrix of the cone into three equal parts. Find the surface area of the cone.

Ball surface area

Cube edge *and.* Find the area of a sphere that passes through all the vertices of one face and touches the parallel face of the cube.

Into a cube whose edge length is *and,* a sphere is inscribed. Find the area of a sphere that touches the inscribed sphere and the three faces of the cube.

Development of the lateral surface of a triangular pyramid. a square with side a. Find the area of a sphere inscribed in this pyramid.

Prove that the area of the spherical surface of the spherical segment *8* 2etLN, where L is the radius of the sphere, and H is the height of the segment.

Height of a regular tetrahedron *H =* 12 cm. The point equidistant from all vertices of the tetrahedron is the center of a sphere with a radius of 4 cm. Determine the area of that part of the sphere that is inside the tetrahedron.

Radii of two balls *and* and *2a.* The center of the smaller ball is on the surface of the larger one. Find the volume and surface area of the common part of these balls.

A regular hexagonal prism is described near the sphere. A plane is drawn through the lateral edge of the prism, which divided the prism into parts with a volume ratio of 1: 5. How are the areas of the parts into which this plane divided the sphere??

A regular triangular prism is described near the sphere. A plane passes through the lateral edge of the prism, which divides the prism into parts with a volume ratio of 1: 2. How are the areas of the parts into which this plane divides the sphere??

## How to cut a pentagon into 4 acute-angled triangles

CUTTING

Cutting polygons in one straight line (levels 1-2).

The tasks in this series can be offered to your child as long as he learns to distinguish between a square and a triangle. You can take scissors and, in his presence, with one movement of your hand, turn a square into two triangles. (It’s good if he is happy about it!). Therefore, the tasks in this section start out very simple. There are no very difficult tasks in this topic, but it allows you to conduct a small scientific research. You can occasionally return to the last tasks “under different sauces” for several years.

Each of these tasks consists of several short ones. Do not rush to ask them all at once, especially if the child is small.

Task 1 Cut a triangle with one line into (a) two triangles; (b) triangle and quadrangle.

The smaller the child, the larger the triangle should be. For little ones, it is good that it be cut out, not drawn: it makes it clearer what an angle is. Require that the children first draw the line with a pencil or pen (you can use a ruler, or you can not; for small children, a ruler.Additional difficulty, and large children should already draw a more or less straight line without it). When the correct drawing is ready, you can, as a reward, really cut the figure with scissors and make sure that all the right angles are in place.

Problem 2 Question for big children: why can’t a pentagon turn out?

Problem 3 Cut a quadrangle (five-, six-) gon with one line into two quadrangles (triangle and pentagon)

There are many such tasks. Naturally, you need to give feasible assignments, at least until your students are ready to distinguish a possible combination from an impossible one. Although, if you ask to cut a square into two pentagons with one line, even kids will probably be indignant. By the way, this problem can also be set with non-convex polygons. the solution is basically the same, but it is more difficult to see it.

When you have gained some experience, you can generalize the problem:

Problem 4 A convex quadrilateral (5-, 6-gon) is cut with one line: what can you get? List all possible combinations of the resulting polygons (and draw the corresponding partitions).

When this problem is solved, it is time to ask the question about the n-gon. But the children, most likely, have not yet grown to algebra and do not understand what it is (even if they already know the word “x”). Therefore, it is better to ask this:

Problem 5 What pairs of polygons can be obtained by cutting a convex 10-gon?

A slightly simpler version of approximately the same task:

Problem 6 Things are not so simple with the corners of polygons: for some reason, when cutting, their total number is not saved. How can it change?

In the previous tasks, we often repeated the word “convex”. Naturally, we did not do this in a conversation with children. if only because defining the concept of bulge is a separate entertainment. To avoid ambiguity, you can always draw and discuss a specific polygon. But sometime you need to remember that polygons are not only convex.

Task 7 Draw a quadrangle and cut it with one line into three parts.

Children taught by the previous tasks are more likely to start proving that this is impossible. The more interesting! If they persist, as a hint, you can ask the question: for which quadrangles the previous results do not work? Maybe it’s some kind of unusual quadrangle?

Another time, you can formulate virtually the same task in a different way:

Problem 8 How to make a hexagon out of a quadrangle with one stroke of scissors?

And it can be even worse. from a 7-gon. a 10-gon And after a while you can put the question like this:

Problem 9 What is the largest-possible-gon that can be obtained from a hexagon by one cut?

You can try to generalize this problem to an arbitrary number of angles.

## How to cut a pentagon into 4 acute-angled triangles

Problem 1: How to cut a 4 × 9 rectangle in two so that they can be folded into a square?

Problem 2: A circle is given and a point inside it is marked. What is the minimum number of parts that this circle can be cut into so that the resulting parts can be folded into a circle in which the marked point is the center.

Problem 3: Cut a corner made up of three cells. into four equal parts.

Problem 4: With the help of cutting and shifting, make a “candy” shape out of the “cross” shape (see picture).

Problem 5: Cut the tetramino into five pieces and add two equal squares from them.

Problem 6: Make four equal rectangles and one square out of a square

a) by cutting and shifting;

b) using only cuts.

Problem 7: a) Is it possible to cut a square into 100 equal non-rectangles quads? b) Is it possible to cut a square into 2000 equal triangles?

Problem 8: Is it possible to make a square of figures? Figures can be borrowed in unlimited quantities. And if the long side of the corner is n cells?

Problem 9: Is it possible to pave the plane with the same a) pentagons; b) hexagons; c) heptagons?

Problem 10: Cut the square into two identical a) pentagons; b) a hexagon; c) 2n-gon; d) 2n 1-gons. Can a rectangle be cut like that? What other figures is this algorithm suitable for??

Problem 11: Is it possible to cut into four acute-angled triangles a) any pentagon; b) regular pentagon?

Problem 12:

The pictures show figures on checkered paper. Your task is to cut each shape into two equal (in shape and size) parts.

(A figure that looks like a rocket must be broken into four identical parts)

## How to cut a pentagon into 4 acute-angled triangles

## Cut correctly into pieces

How a given rectangle should be cut into two parts so that they can be folded: 1) a triangle, 2) a parallelogram (other than a rectangle), 3) a trapezoid?

A rectangle is given, the base of which is twice the height. 1) How should this rectangle be cut into two parts so that they can be used to make an isosceles triangle? 2) How to cut a given rectangle into three parts, from which one could make a square?

How can an equilateral triangle be cut into: 1) two equal triangles; 2) three equal triangles; 3) four equal triangles; 4) six equal triangles; 5) eight; 6) twelve?

Two equal squares are given. How to cut each of them into two parts so that the resulting parts can be folded into a square?

How to divide a given rectangle by two rectilinear cuts into two equal pentagons and two equal right-angled triangles?

Two unequal squares are given. How should they be cut into pieces so that a third square can be folded? How is the side length of the third square expressed in terms of the side lengths of the two data?

A rectangular chocolate bar consists of mn single square slices. How many faults have to be made (one piece breaks at a time) to break this tile into single square pieces?

How many cuts should be made with planes so that from a cube with an edge of 3 dm we get cubes with an edge of 1 dm?

A right-angled triangle is given. How to cut it into two such parts so that from them (without turning it over with the back side) you can fold a triangle symmetrical to this one with respect to one of its legs?

Given triangle ABC. How to cut it into pieces so that from them (without turning the back side) you can fold a triangle symmetrical to this one with respect to the base of the speaker?

Cut the square into pieces, as shown in Figure 49, mix them and then fold: 1) the same square; 2) a right-angled isosceles triangle; 3) a rectangle other than a square; 4) a parallelogram other than a rectangle; 5) trapezoid.

A painted cube with an edge of 10 cm was sawed into cubes with a rhombus of 1 cm. How many cubes will be obtained: 1) with one painted edge; 2) with two; 3) with three; 4) have no painted edges at all?

How to cut a rectangle with sides 16 and 9 cm into two parts so that a square can be folded out of them? (The cut may be in the form of a broken mowing line.)

Copy each of the shapes in Figure 50 and cut it into 4 equal parts.

1) How to cut a given right-angled triangle into acute-angled triangles? 2) How to cut a given arbitrary triangle into acute-angled triangles?

Inside the convex column, 10 points are marked, of which no three lie on one straight line. The polygon is cut into triangles so that all the vertices of the given column and all the given ten points serve as their vertices. How many triangles will there be?

## Abstract of a lesson in mathematics “Triangles

EMC “Planet of Knowledge”

Mathematics Grade 2

Topic: “TRIANGLES”

The goals of the teacher: to acquaint with the types of triangles (rectangular, acute-angled, obtuse-angled, equilateral); promote the development of the ability to distinguish between the types of triangles in the drawing, determine the area of figures by cells, cut a rectangle into two triangles.

Planned educational outcomes.

Subject: they know how to distinguish between right, acute and obtuse angles in drawings, recognize a right-angled triangle, determine the area of a rectangle (in conventional units based on the illustration).

Metasubject (criteria for the formation / assessment of the components of universal educational actions. UUD): regulatory: organize a mutual review of the work performed; plan their own computing activities; cognitive: experimenting with triangles (the number of right and obtuse angles); perform calculations by analogy; calculate the area of a polygonal figure by dividing it into rectangles; construct figures from parts of a rectangle; communicative: answering questions, asking questions, clarifying incomprehensible.

During the classes

I. Updating knowledge. Verbal counting.

The mole built a triangular fence around the house. The length of the smallest side of the fence is 3 cm, the largest is 5 cm.What is the third side if the perimeter of the triangle is 12 cm?

In a quadrilateral, draw 2 line segments so that it divides:

3 triangles; 4 triangles.

Count how many triangles are in each picture.

II. Lesson topic message.

III. Learning new material.

Textbook work.

Task 1. Look at the pictures. What unites them? Read the text of the tutorial. What is “shamrock”, “trilogy”, “triukh”, “triptych”, “triangle”.

Task 2. Name the types of corners. (Straight, obtuse, acute.) What is the name of a triangle in which there is a right angle? (Right triangle.)

Task 3. Try to draw a triangle with two right angles. (They try.) Is it possible? (Not.)

findings.

In a right-angled triangle, one corner of a straight line and two acute angles.

An obtuse triangle has one obtuse angle and two acute angles.

In an acute-angled triangle, all corners are sharp.

Task 4. Measure the sides of these triangles. What have you noticed? (These triangles have all sides equal.)

A triangle in which all sides are equal is called equilateral.

Output. All equilateral triangles have the same shape, but may have different sizes.

Task 5. The rectangle was cut into two identical right-angled triangles. What shapes can be added from these triangles? (Larger triangles.)

Task 6. How many cells do the houses occupy together with the roof? (12; 12; 15; 11; 14; 15.)

Task 7 (work in pairs).

IV. Practical frontal work.

Look at the shapes and find the “extra”. What is the common feature of the remaining shapes.

(The extra one is a rectangle. All the remaining figures are open.)

(The extra one is a circle. The rest of the figures have sides and corners.)

(Superfluous. hours. Others. geometric shapes.)

Graphic dictation: 1 class. to the right, 2 cl. to the right upwards diagonally, 2 cl. left, 2 cl. right down diagonally, 1 cl. right, etc. Continue the pattern to the end of the line.

A group of triangles on the board. After the children’s answer, a sign “Equilateral” is attached.

3 pupils in “triangle” hats come to the blackboard and read:

You look at me, you look at him, look at all of us:

We have everything, we have everything, we only have three!

Three sides and three corners and the same number of vertices,

And three times difficult things we will do three times.

Everyone in our city is friends, you can’t find a friendlier one.

We are a family of triangles, everyone should know us!

V. Lesson summary.

Vi. Reflection

### View document content

“Summary of the lesson in mathematics” Triangles “”

EMC “Planet of Knowledge”

Mathematics Grade 2

Topic: “TRIANGLES”

The goals of the teacher: to acquaint with the types of triangles (rectangular, acute-angled, obtuse-angled, equilateral); promote the development of the ability to distinguish between the types of triangles in the drawing, determine the area of figures by cells, cut a rectangle into two triangles.

Planned educational outcomes.

Subject: *know how* distinguish between right, acute and obtuse angles in figures, recognize a right-angled triangle, determine the area of a rectangle (in conventional units based on the illustration).

Metasubject (criteria for the formation / assessment of the components of universal educational actions. UUD): *regulatory:* organize a mutual review of the work performed; plan their own computing activities; *cognitive:* experimenting with triangles (the number of right and obtuse angles); perform calculations by analogy; calculate the area of a polygonal figure by dividing it into rectangles; construct figures from parts of a rectangle; *communicative:* answer questions, ask questions, clarify incomprehensible.

I. Updating knowledge. Verbal counting.

The mole built a triangular fence around the house. The length of the smallest side of the fence is 3 cm, the largest is 5 cm.What is the third side if the perimeter of the triangle is 12 cm?

In a quadrilateral, draw 2 line segments so that it divides:

3 triangles; 4 triangles.

Count how many triangles are in each picture.

II. Lesson topic message.

III. Learning new material.

Textbook work.

Task 1. Look at the pictures. What unites them? Read the text of the tutorial. What is “shamrock”, “trilogy”, “triukh”, “triptych”, “triangle”.

Task 2. Name the types of corners. *(Straight, blunt, sharp.)* What is the name of a triangle in which there is a right angle? *(Right triangle.)*

Task 3. Try to draw a triangle with two right angles. *(They try.)* Is it possible? *(Not.)*

findings.

*In a right-angled triangle, one corner of a straight line and two acute angles.*

*An obtuse triangle has one obtuse angle and two acute angles.*

*In an acute-angled triangle, all corners are sharp.*

Task 4. Measure the sides of these triangles. What did you notice? *(These triangles have all sides equal.)*

*A triangle in which all sides are equal is called equilateral.*

Output. *All equilateral triangles have the same shape, but may have different sizes.*

Task 5. The rectangle was cut into two identical right-angled triangles. What shapes can be folded from these triangles? *(Larger triangles.)*

Task 6. How many cells do the houses occupy with the roof?? *(12; 12; 15; 11; 14; 15.)*

Task 7 (work in pairs).

IV. Practical frontal work.

Look at the shapes and find the “extra”. What is the common feature of the remaining shapes.

*(The extra one is a rectangle. All the remaining figures are open.)*

*(The extra one is a circle. The rest of the figures have sides and corners.)*

*(Superfluous. hours. Others. geometric shapes.)*

Graphic dictation: 1 class. to the right, 2 cl. to the right upwards diagonally, 2 cl. left, 2 cl. right down diagonally, 1 cl. right, etc. Continue the pattern to the end of the line.

*A group of triangles on the board. After the children’s answer, a sign “Equilateral” is attached.*

*3 pupils in “triangle” hats come to the blackboard and read:*

You look at me, you look at him, look at all of us:

We have everything, we have everything, we only have three!

Three sides and three corners and the same number of vertices,

And three times difficult things we will do three times.

Everyone in our city is friends, you can’t find a friendlier one.

We are a family of triangles, everyone should know us!

V. Lesson summary.

## Regular pentagon

By the theorem on the sum of the angles of a convex polygon, the sum of the angles of a regular pentagon is 180º (5-2) = 540º.

Since all angles of a regular n-gon are equal to each other, each inner corner of a regular pentagon is 540º: 5 = 108º (in particular, ∠A2A1A5 = 108º).

The sum of the outside angles of the polygon, taken one at each vertex, is 360º. Since all external angles of a regular pentagon are equal to each other, the degree measure of each, for example, angle 1, is equal to

∠1 = 360º: 5 = 72º (it was possible to search for the outer corner as adjacent to the inner one).

Each central corner of a regular pentagon, for example A1O A2, is

Like any other regular polygon, a regular pentagon is inscribed in a circle and circumscribed about a circle.

Connecting the center of a regular polygon with its vertices, we get five equal isosceles triangles.

The base of each such triangle is equal to the side of the 5-gon, the sides are equal to the radius of the circumscribed circle, the angle at the apex is equal to the central angle of the 5-gon.

Draw the height OF from the vertex.

By the property of an isosceles triangle, OF is also the median and bisector of triangle A1OA5, that is

Consider right triangle A1OF.

Thus, the formula for the radius of a circle circumscribed about a regular pentagon is

Substituting the value of the cotangent 36 °, we get:

So, the formula for the radius of a circle inscribed in a regular pentagon is

you can find the area of a regular pentagon. Here

All diagonals of a regular pentagon are equal.

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