Sides AB and BC of triangle ABC are respectively 15 and 20, height BD is 12. Find the length of side AC.

1. Consider a right-angled triangle ABD. By the Pythagorean theorem (the square of the hypotenuse is equal to the sum of the squares of the legs):

AD ^ 2 + BD ^ 2 = AB ^ 2:

AD ^ 2 + 12 ^ 2 = 15 ^ 2;

AD ^ 2 + 144 = 225;

AD ^ 2 = 225 – 144;

AD ^ 2 = 81;

AD = 9.

2. Consider a right-angled triangle CBD. By the Pythagorean theorem (the square of the hypotenuse is equal to the sum of the squares of the legs):

CD ^ 2 + BD ^ 2 = BC ^ 2;

CD ^ 2 + 12 ^ 2 = 20 ^ 2;

CD ^ 2 + 144 = 400;

cD ^ 2 = 400 – 144;

CD ^ 2 = 256;

CD = 16.

3. AC = AD + DC;

AC = 9 + 16;

AC = 25.

Answer: 25.