Review questions for Chapter 7. Review questions for Chapter VI.

1. Explain how the areas of polygons are measured.

2. Formulate the basic properties of the areas of polygons.

3. Which polygons are called equal-sized and which are called equi-contiguous?

4. Formulate and prove a theorem about calculating the area of a rectangle.

5. Formulate and prove a theorem on calculating the area of a parallelogram.

6. Formulate and prove a theorem about calculating the area of a triangle. How to calculate area right triangle on his legs?

7. Formulate and prove a theorem about the ratio of the areas of two triangles that have equal angles.

8. Formulate and prove a theorem about calculating the area of a trapezoid.

9. Formulate and prove the Pythagorean theorem.

10. Formulate and prove the theorem converse to the Pythagorean theorem.

11. What triangles are called Pythagorean? Give examples of Pythagorean triangles.

12. What formula for the area of a triangle is called Heron’s formula? Derive this formula.

Additional tasks

500. Prove that the area of a square constructed on the side of an isosceles right triangle is twice more area a square constructed at a height drawn to the hypotenuse.

501. Area land plot equal to 27 hectares. Express the area of the same area: a) in square meters; b) in square kilometers.

502. The heights of the parallelogram are 5 cm and 4 cm, and the perimeter is 42 cm. Find the area of the parallelogram.

503. Find the perimeter of a parallelogram if its area is 24 cm 2 and the point of intersection of the diagonals is 2 cm and 3 cm away from the sides.

504. The smaller side of the parallelogram is 29 cm. A perpendicular drawn from the point of intersection of the diagonals to the larger side divides it into segments equal to 33 cm and 12 cm. Find the area of the parallelogram.

505. Prove that of all triangles in which one side is equal to a and the other is b, largest area has one whose sides are perpendicular.

506. How to draw two straight lines through the vertex of a square in order to divide it into three figures whose areas are equal?

507.* Each side of one triangle is greater than any side of the other triangle. Does it follow from this that the area of the first triangle is greater than the area of the second triangle?

508.* Prove that the sum of distances from a point based on isosceles triangle to the sides does not depend on the position of this point.

509. Prove that the sum of the distances from a point lying inside an equilateral triangle to its sides does not depend on the position of this point.

510.* Through point D, lying on side BC of triangle ABC, lines are drawn parallel to the other two sides and intersecting sides AB and AC, respectively, at points E and F. Prove that triangles CDE and BDF are equal in size.

511. In a trapezoid ABCD with sides AB and CD, the diagonals intersect at point O.

a) Compare the areas of triangles ABD and ACD.
b) Compare the areas of triangles ABO and CDO.
c) Prove that the equality OA OB = OS OD holds.

512.* The bases of a trapezoid are equal to a and b. A segment with ends on the sides of the trapezoid, parallel to the bases, divides the trapezoid into two equal trapezoids. Find the length of this segment.

513. The diagonals of a rhombus are 18 m and 24 m. Find the perimeter of the rhombus and the distance between parallel sides.

514. The area of a rhombus is 540 cm 2, and one of its diagonals is 4.5 dm. Find the distance from the point of intersection of the diagonals to the side of the rhombus.

515. Find the area of an isosceles triangle if: a) the side is 20 cm and the angle at the base is 30°; b) the height drawn to the side is 6 cm and forms an angle of 45° with the base.

516. In triangle ABC, BC = 34 cm. The perpendicular MN drawn from the middle of BC to the straight line AC divides side AC into segments AN = 25 cm and NC = 15 cm. Find the area of triangle ABC.

517. Find the area of quadrilateral ABCD, in which AB = 5 cm, BC = 13 cm, CD = 9 cm, DA = 15 cm, AC = 12 cm.

518. Find the area of an isosceles trapezoid if: a) its smaller base is 18 cm, its height is 9 cm and its acute angle is 45°; b) its bases are 16 cm and 30 cm, and its diagonals are mutually perpendicular.

519. Find the area of an isosceles trapezoid whose height is equal to h and whose diagonals are mutually perpendicular.

520. The diagonals of an isosceles trapezoid are mutually perpendicular, and the sum of the bases is 2a. Find the area of the trapezoid.

521. Prove that if the diagonals of the quadrilateral ABCD are mutually perpendicular, then AD 2 + BC 2 = AB 2 + CD 2.

522. In an isosceles trapezoid ABCD with bases AD = 17 cm, BC = 5 cm and side AB = 10 cm, a straight line is drawn through vertex B, bisecting diagonal AC and intersecting base AD at point M. Find the area of triangle BDM.

523. Two squares with side a have one common vertex, and the side of one of them lies on the diagonal of the other. Find the area of the common part of these squares.

524. The sides of the triangle are 13 cm, 5 cm and 12 cm. Find the area of this triangle.

525. The distance from point M, lying inside triangle ABC, to line AB is 6 cm, and to line AC is 2 cm. Find the distance from point M to line BC, if AB = 13 cm, BC = 14 cm, AC = 15 cm .

526. In a rhombus, the height equal to cm is 2/3 of the larger diagonal. Find the area of the rhombus.

527. In an isosceles trapezoid, the diagonal is 10 cm and the height is 6 cm. Find the area of the trapezoid.

528. In trapezoid ABCD, the diagonals intersect at point O. Find the area of triangle AOB if the lateral side CD of the trapezoid is 12 cm, and the distance from point O to the straight line CD is 5 cm.

529. The diagonals of a quadrilateral are 16 cm and 20 cm and intersect at an angle of 30°. Find the area of this quadrilateral.

530. In an isosceles triangle ABC with base BC, the height AD is 8 cm. Find the area of triangle ABC if the median DM of triangle ADC is 8 cm.

531. Sides AB and BC of rectangle ABCD are equal to 6 cm and 8 cm, respectively. A line passing through vertex C and perpendicular to line BD intersects side AD at point M, and diagonal BD at point K. Find the area of quadrilateral ABKM.

532. In triangle ABC, the height BH is drawn. Prove that if:

a) angle A is acute, then BC 2 = AB 2 + AC 2 - 2AC AN;
b) angle A is obtuse, then BC 2 = AB 2 + AC 2 + 2AC AN.

Answers to problems

1. How many lines can be drawn through two points?

2. How many common points can two straight lines have?

3. Explain what a segment is.

4. Explain what a beam is. How are the rays designated?

5. What figure is called an angle? Explain what the vertex and sides of an angle are.

6. Which angle is called unfolded?

7. What figures are called equal?

8. Explain how to compare two line segments.

9. Which point is called the midpoint of the segment?

10. Explain how to compare two angles.

11. Which ray is called the bisector of an angle?

12. Point C divides segment AB into two segments. How to find the length of segment AB if the lengths of segments AC and CB are known?

13. What tools are used to measure distances?

14. What is the degree measure of an angle?

15. Ray OS divides angle AOB into two angles. How to find the degree measure of angle AOB if you know degree measures corners AOS and SOV?

16. Which angle is called acute? straight? stupid?

17. What angles are called adjacent? What is the sum of adjacent angles?

18. What angles are called vertical? What properties do vertical angles have?

19. Which lines are called perpendicular?

20. Explain why two lines perpendicular to the third do not intersect.

21. What devices are used to construct right angles on the ground?

Additional tasks for Chapter I

71. Mark four points so that no three lie on the same straight line. Draw a straight line through each pair of points. How many straight lines did you get?

72. Given four lines, every two of which intersect. How many points of intersection do these lines have if only two lines pass through each point of intersection?

73. How many undeveloped angles are formed when three lines passing through one point intersect?

74. Point N lies on the segment MP. The distance between points M and P is 24 cm, and the distance between points N and M is twice the distance between points N and P. Find the distance:

a) between points N and P;
b) between points N and M.

75. Three points K, L, M lie on the same straight line, KL = 6 cm, LM = 10 cm. What could be the distance KM? For each of the possible cases, make a drawing.

76. A segment AB of length a is divided by points P and Q into three segments AP, PQ and QB so that AP - 2PQ = 2QB. Find the distance between:

a) point A and the middle of the segment QB;
b) the midpoints of segments AP and QB.

77. A segment of length m is divided:

a) into three equal parts;
b) into five equal parts.

Find the distance between the middles of the extreme parts.

78. A segment of 36 cm is divided into four unequal parts. The distance between the centers of the extreme parts is 30 cm. Find the distance between the centers of the middle parts.

79. Points A, B and C lie on the same line, points M and N are the midpoints of segments AB and AC. Prove that BC = 2MN.

80. It is known that ZAOB = 35°, ZBOC = 50°. Find the angle AOC. For each possible case, make a drawing using a ruler and protractor.

81. Angle hk is equal to 120°, and angle hm is equal to 150°. Find the angle km. For each of the possible cases, make a drawing.

82. Find adjacent angles if:

a) one of them is 45° larger than the other;
b) their difference is 35°.

83. Find the angle formed by the bisectors of two adjacent angles.

84. Prove that the bisectors of vertical angles lie on the same straight line.

85. Prove that if the bisectors of angles ABC and CBD are perpendicular, then points A, B and D lie on the same straight line.

86. Given two intersecting lines a and b and a point A not lying on these lines. Lines m and n are drawn through point A so that m⊥a, n⊥b. Prove that the lines m and n are not the same.

Ready-made homework for a geometry textbook for students in grades 7-9, authors: L.S. Atanasyan, V.F. Butuzov, S.B. Kadomtsev, E.G. Poznyak, I.I. Yudina, Prosveshchenie publishing house for the 2015-2016 academic year.

Guys, in grades 7-9 you will study such an interesting subject as geometry. To avoid having problems understanding this lesson in the future, you need to work hard from the very beginning.

In previous classes you have already become acquainted with some geometric shapes. In this buzz you will expand this minimum of knowledge. The entire course is divided into two sections: planimetry and stereometry. In grades 7 and 8 you will look at figures on a plane - this is a section on planimetry. In 9th grade, properties of figures in space - stereometry.

Often a situation arises when it is not possible, based on the condition, to do correct drawing, draw all the details in space and then geometry seems like an overwhelming subject for you. If you start having such difficulties, then we recommend using our geometry test for grades 7-9 L.S. Atanasyan, which is posted below.

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1. What is the ratio of two segments called?

2. In what case are they said that segments AB and CD are proportional to segments A 1 B 1 and C 1 D 1?

3. Define similar triangles.

4. Formulate and prove a theorem on the ratio of the areas of similar triangles.

5. Formulate and prove a theorem expressing the first sign of similarity of triangles.

6. Formulate and prove a theorem expressing the second criterion for the similarity of triangles.

7. Formulate and prove a theorem expressing the third criterion for the similarity of triangles.

8. Which segment is called the midline of the triangle? State and prove the theorem about the midline of a triangle.

9. Prove that the medians of a triangle intersect at one point, which divides each median in a 2:1 ratio, counting from the vertex.

10. Formulate and prove the statement that the altitude of a right triangle drawn from the vertex right angle, divides a triangle into similar triangles.

11. State and prove statements about proportional segments in a right triangle.

12. Give an example of solving a construction problem using the similarity method.

13. Tell us how to determine the height of an object on the ground and the distance to an inaccessible point.

14. Explain which two figures are called similar. What is the similarity coefficient of figures?

15. What is called the sine, cosine, tangent of an acute angle of a right triangle?

16. Prove that if the acute angle of one right triangle is equal to sharp corner another right triangle, then the sines of these angles are equal, the cosines of these angles are equal and the tangents of these angles are equal.

17. What equality is called the basic trigonometric identity?

18. What are the values of sine, cosine and tangent for angles of 30°, 45°, 60°? Justify your answer.

Additional tasks

604. Triangles ABC and A 1 B 1 C 1 are similar, AB = 6 cm, BC - 9 cm, C A = 10 cm. The largest side of triangle A 1 B 1 C 1 is 7.5 cm. Find the other two sides of triangle A 1 B 1 C 1 .

605. Diagonal AC of the trapezoid ABCD divides it into two similar triangles. Prove that AC 2 = a b, where a and b are the bases of the trapezoid.

606. Bisectors MD and NK of triangle MNP intersect at point O. Find the relation OK: ON if MN = 5 cm, NP = 3 cm, MP = 7 cm.

607. The base of an isosceles triangle is related to the side as 4: 3, and the height drawn to the base is 30 cm. Find the segments into which the bisector of the angle at the base divides this height.

608. On the continuation of the lateral side OB of the isosceles triangle AO B with base AB, point C is taken so that point B lies between points O and C. Segment AC intersects the bisector of angle AOB at point M. Prove that AM< МС.

609. Point D is taken on side BC of triangle ABC so that Prove that AD is the bisector of triangle ABC.

610. A straight line parallel to side AB of triangle ABC divides side AC in the ratio 2:7, counting from vertex A. Find the sides of the cut triangle if AB = 10 cm, BC = 18 cm, CA = 21.6 cm.

611. Prove that the median AM of triangle ABC bisects any segment parallel to side BC whose ends lie on sides AB and AC.

612. Two poles AB and CD different lengths a and b are installed vertically at a certain distance from each other as shown in Figure 210. Ends A and D, B and C are connected by ropes that intersect at point O. Based on the figure, prove that:

Find x and prove that x does not depend on the distance d between poles AB and CD.

Rice. 210

613. Prove that triangles ABC and A 1 B 1 C 1 are similar if:

A) , where VM and B 1 M 1 are the medians of the triangles;

b) ∠A = ∠A 1, , where ВН and В 1 Н 1 are the heights of triangles АВС and A 1 B 1 C 1.

614. The diagonals of a rectangular trapezoid ABCD with right angle A are mutually perpendicular. The base AB is 6 cm and the side AD is 4 cm. Find DC, DB and CB.

615.* A segment with ends on the sides of a trapezoid is parallel to its bases and passes through the point of intersection of the diagonals. Find the length of this segment if the bases of the trapezoid are equal to a and b.

616. Prove that the vertices of a triangle are equidistant from the line containing its midline.

617. Prove that the midpoints of the sides of a rhombus are the vertices of a rectangle.

618. Points M and N are respectively the midpoints of sides CD and BC of parallelogram ABCD. Prove that the lines AM and AN divide the diagonal BD into three equal parts.

619. The bisector of the exterior angle at vertex A of triangle ABC intersects line BC at point D. Prove that .

620. In triangle ABC (AB≠ AC), a line is drawn through the middle of side BC, parallel to the bisector of angle A, which intersects lines AB and AC, respectively, at points D and E. Prove that BD = CE.

621. In a trapezoid ABCD with bases AD and BC, the sum of the bases is b, the diagonal AC is a, ∠ACB = α. Find the area of the trapezoid.

622. Point K is marked on side AD of parallelogram ABCD such that AK = 1/4 KD. Diagonal AC and segment B K intersect at point P. Find the area of parallelogram ABCD if the area of triangle ARK is 1 cm 2.

623. In a rectangular trapezoid ABCD with bases AD and BC ∠A = ∠B = 90°, ∠ACD = 90°, BC = 4 cm, AD = 16 cm. Find angles C and D of the trapezoid.

624. Prove that the medians of a triangle divide it into six triangles whose areas are pairwise equal.

625. The base AD of an isosceles trapezoid ABCD is 5 times larger than the base BC. The height BH intersects the diagonal AC at point M, the area of the triangle AMN is 4 cm 2. Find the area of trapezoid ABCD.

626. Prove that triangles ABC and A 1 B 1 C 1 are similar if where AD and A 1 D 1 are the bisectors of the triangles.

Construction tasks

627. Given a triangle ABC. Construct a triangle A1B1C1, similar to triangle ABC, the area of which is twice the area of triangle ABC.

628. Given three segments, the lengths of which are respectively equal to a, b and c. Construct a segment whose length is equal to .

629. Construct a triangle if the midpoints of its sides are given.

630. Construct a triangle using a side and medians drawn to the other two sides.

Answers to problems

1 Give examples of vector quantities known to you from your physics course.

2 Define a vector. Explain which vector is called zero.

3 What is the length of a non-zero vector called? What is the length of the zero vector?

4 What vectors are called collinear? Draw in the figure co-directional vectors and and oppositely directed vectors

5 Define equal vectors.

6 Explain the meaning of the expression: “The vector is delayed from point A.” Prove that from any point you can plot a vector equal to the given one, and only one.

7 Explain what vector is called the sum of two vectors. What is the triangle rule for adding two vectors?

8 Prove that for any vector the equality

9 Formulate and prove a theorem about the laws of vector addition.

10 What is the parallelogram rule for adding two non-collinear vectors?

11 What is the polygon rule for adding several vectors?

12 What vector is called the difference of two vectors? Construct the difference of two given vectors.

13 Which vector is called opposite to this one? Formulate and prove the vector difference theorem.

14 What vector is called the product of a given vector and a given number?

15 What is the product equal to

16 Can vectors be non-collinear?

17 Formulate the basic properties of multiplying a vector by a number.

18 Give an example of using vectors to solve geometric problems.

19 Which segment is called the midline of the trapezoid?

20 State and prove the theorem about the midline of a trapezoid.

Additional tasks for Chapter IX

800. Prove that if the vectors are co-directional, then and if they are oppositely directed, and then

801. Prove that the inequalities are valid for any vectors

802. On side BC of triangle ABC, point N is marked so that BN = 2NC. Express vector in terms of vectors

803. On sides MN and NP of triangle MNP points X and Y are marked respectively so that

804. The base AD of the trapezoid ABCD is three times larger than the base BC. On side AD there is a point K such that Express vectors in terms of vectors