In an isosceles triangle ABC, base AC = 30, height BH = 20. Find the AK height.

The height of ВН is the median of the triangle ABC, since it is isosceles, then AH = CH = AС / 2 = 30/2 = 15 cm.

In a right-angled triangle ВСН, we define, by the Pythagorean theorem, the length of the hypotenuse ВС.

BC ^ 2 = BH ^ 2 + CH ^ 2 = 400 + 225 = 625.

BC = 25 cm.

Let us determine the area of the triangle ABC through the height of the HВ and the base of the AC.

Savs = AC * ВН / 2 = 30 * 20/2 = 300 cm2.

Let us determine the area of the triangle ABC through the height of the AK and the side BC.

Savs = BC * AK / 2 = AK * 25/2.

Then AK * 25/2 = 300.

AK = 2 * 300/25 = 24 cm.

Answer: The length of the AK height is 24 cm.