1. On the line there are points A, B and C. Find the length of segment MK, where M is the midpoint of segment AB, K is the midpoint of BC, and AB = 50 cm, BC = 16 cm.

2. Find the lengths of segments OB and OA if AB = 36 cm, and OB is 3 times shorter than OA.

3. Three rays AM, AN and AK are drawn from point A. Find angle NAK if ∠MAN = 76°, ∠MAK = 36°.

4. In an isosceles triangle with a perimeter of 48 cm, the side refers to the base as 5 : 2. Find the sides of the triangle.

5. In isosceles triangle ABC the points K and M are the midpoints of the sides AB and BC, respectively. BD is the median of triangle. Prove that ΔBKD = ΔBMD.

6. Construct diameter AB and equal chords AC and AD in the circle. Prove that Δ ABC = Δ ABD.

**Variant 1**

1. Three points M, N and K lie on one line. It is known that MN = 15 cm, NK = 18 cm. What can be the distance MK?

2. The sum of the vertical angles MOE and DOC formed by intersecting the lines MC and DE is 204°. Find the angle MOD.

3. From point B there are three rays: BM, BN and BK. Find angle NBK if ∠MBN = 84°, ∠MBK = 22°.

4. In an isosceles triangle with perimeter 72 cm, the base refers to the side as 2 : 5. Find the sides of the triangle.

5. The segments AB and MK intersect at point O, which is the midpoint of the segment MK, ∠BMO = ∠AKO. Prove that Δ MOW = Δ KOA.

6. In the circle with center O there are chords DE and RK, and ∠DOE = ∠POK. Prove that these chords are equal.

**Variant 2**

1. Three points B, C and D lie on the same line. It is known that BD = 17 cm, DC = 25 cm. What could be the length of segment BC?

2. The sum of the vertical angles AOB and SBD formed by intersecting the lines AD and BC is 108°. Find the angle ABD.

3. From point M there are three rays: MO, MN and MK. What is the angle NMK if ∠OMN = 78°, ∠OMK = 30°?

4. In an isosceles triangle with a perimeter of 56 cm, the base refers to the side as 2 : 3. Find the sides of the triangle.

5. The segments AB and MK intersect at point O, which is the midpoint of the segment MK, ∠BMO = ∠AKO. Prove that Δ MOW = Δ KOA.

6. In a circle with center O the diameters MK and PH are drawn, and ∠ ORK = 40°. Find ∠OMH.