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## Wednesday - Geometry in the plane

1. The center of a circle inscribed in a rectangular trapezoid is at a distance of 3 and 9 cm from the ends of the lateral side. Find the sides of the trapezoid.
Answer: $\frac{36}{\sqrt10}$, $\frac{12}{\sqrt10}$, $\frac{18}{\sqrt10}$, $\frac{30}{\sqrt10}$ cm

2.  Inside the right angle, a point M is given, the distances from which to the sides of the angle are 4 and 8 cm. The straight line passing through point M cuts off a triangle with an area of 100 $cm^2$ from the right angle. Find the legs of a triangle.
Answer: 20,10 or 5,40 𝑐𝑚

3. Point 𝑪𝟏 is the midpoint of side 𝑨𝑩 of triangle 𝑨𝑩𝑪; angle 𝑪𝑶𝑪𝟏, where 𝑶 the center of a circle circumscribed about a triangle is right. Prove that |𝒁𝑩 – 𝒁𝑨| = 𝟗𝟎°.

4. The circle touches two adjacent sides of the square and divides each of its two other sides into segments equal to 2 and 23 cm. Find the radius of the circle.
Answer: 17cm.

5. Given a triangle ABC, in which 2hс = AB and ZA = 75°. Find the value of the angle C.
Answer: 75

6. The bisectors of obtuse angles at the base of the trapezoid intersect at its other base. Find the sides of a trapezoid if its height is 12 cm and the lengths of the bisectors are 15 and 13 cm.
Answer: 14, 12.5, 29.4, 16.9

7. Given a triangle ABC such that AB = 15 cm, BC = 12 cm and AC = 18 cm. In what ratio does the center of the circle inscribed in the triangle divide the bisector of angle C?
Answer: $2:1$

8. Two chords 9 and 17 cm long are drawn from one point of the circle. Find the radius of the circle if the distance between the midpoints of these chords is 5 cm.
Answer: $\frac{85}{8}$

9. A circle with a radius of 5 cm is inscribed at a certain angle. The length of the chord connecting the points of tangency is 8 cm. Two tangents are drawn to the circle, parallel to the chord. Find the sides of the resulting trapezoid.
Answer: 5, 20, 12.5, 12.5

10. The vertices of a rectangle inscribed in a circle divide it into four arcs. Find the distance from the middle of one of the larger arcs to the vertices of a rectangle if its sides are 24 and 7 cm.
Answer: 15 and 20 cm

11. The center of a semicircle inscribed in a right-angled triangle so that its diameter lies on the hypotenuse divides the hypotenuse into segments 30 and 40. Find the length of the arc of the semicircle enclosed between the points of its tangency with legs.
Answer: $12\pi$

12. In triangle ABC, the value of angle A is twice the value of angle B, and the lengths of the sides opposite to these angles are 12 and 8 cm, respectively. Find the length of the third side of the triangle.
Answer: 10 cm

13.  The largest of the parallel sides of the trapezoid is a, the smaller is b, the non-parallel sides are c and d. Find the area of the trapezoid.
Answer: $\frac{a+b}{4(a-b)}\sqrt{(𝑎−𝑏+𝑐+𝑑)(𝑎−𝑏+𝑑−𝑐)(𝑐+𝑎−𝑏−𝑑)(𝑐−𝑎+𝑏+𝑑)}$

14. The height and median of a triangle, drawn inside it from one of its vertices, are different and form equal angles with the sides extending from the same vertex. Determine the radius of the circumscribed circle if the median is m.
Answer: m