Find the height of an isosceles triangle whose side and base are 13 and 10, respectively.

Given:
ABC – isosceles triangle,
AB = BC = 13,
AC = 10,
ВН – the height of the isosceles triangle ABC.
Find the length of the height of the HВ -?
Decision:
1. Consider an isosceles triangle ABC. The HL height is also the median. Therefore AH = HC = 10: 2 = 5.
2. Then by the Pythagorean theorem (the square of the hypotenuse is equal to the sum of the squares of the legs):
HC ^ 2 + BH ^ 2 = BC ^ 2:
BH ^ 2 = BC ^ 2 – HC ^ 2;
BH ^ 2 = 13 ^ 2 – 5 ^ 2;
BH ^ 2 = 13 * 13 – 5 * 5;
BH ^ 2 = 169 – 25;
BH ^ 2 = 144;
BH = 12.
Answer: the length of the height HH = 12.