In an isosceles triangle ABC, the median BK = 10, the lateral side BC = 26.

In an isosceles triangle ABC, the median BK = 10, the lateral side BC = 26. Find the line segment MN if you know that it connects the midpoints of the sides.

If triangle ABC is isosceles, then the median BK is also the height. In a right-angled triangle ABK, the leg AK, according to the Pythagorean theorem, is connected to the other sides of this triangle by the formula AB ^ 2 = AK ^ 2 + ВK ^ 2.

Whence AK = (26 ^ 2 – 10 ^ 2) ^ (1/2) = 24.

The base of the triangle ABC is equal to AC = 2AK = 2 * 24 = 48.

The sought segment MN connects the midpoints of the lateral sides, which means it is the middle line of the triangle ABC, which is equal to half the base of the AC:

MN = AC / 2 = 48/2 = 24.

Answer: the segment MN is 24.