In an isosceles triangle ABC AC = BC / find AC if the height is AH = 8 AB = 10.

In an isosceles triangle ABC, it is known:

AC = BC;
Height AH = 8;
AB = 10.
Find AC of triangle ABC.

Solution:

1) The height AH is perpendicular to the BC side.

2) Consider triangle АНВ with right angle H.

The leg and hypotenuse are known. Let’s find the second leg according to the Pythagorean theorem.

BH = √ (10 ^ 2 – 8 ^ 2) = √ (100 – 64) = √36 = √6 ^ 2 = 6;

3) Find the side of the CH.

AH ^ 2 = BH * CH;

8 ^ 2 = 6 * CH;

64 = 6 * CH;

CH = 64/6 = 60/6 + 4/6 = 10 + 4/6 = 10 + 2/3 = 10 2/3;

4) Find the side of the BC.

BC = CH + BH = 10 2/3 + 6 = 10 + 2/3 + 6 = 16 + 2/3 = 16 2/3;

5) BC = AC = 16 2/3.

Answer: AC = 16 2/3.