The legs of a right-angled triangle are 12: 5. The hypotenuse is 39 cm. Find the sides of the triangle.

Given:
right-angled triangle ABC,
angle С = 90 degrees,
AC: BC = 12: 5,
AB = 39 centimeters.
Find legs AC, BC -?
Decision:
Consider a right-angled triangle ABC. Let the length of the leg AC = 12 * x centimeters, and the length of the leg BC = 5 * x centimeters. Then, by the Pythagorean theorem (the square of the hypotenuse is equal to the sum of the squares of the legs):
AC ^ 2 + BC ^ 2 = AB ^ 2:
(12x) ^ 2 + (5x) ^ 2 = 39 ^ 2;
144x ^ 2 + 25 x ^ 2 = 1521;
169x ^ 2 = 1 521;
x ^ 2 = 1 521: 169;
x ^ 2 = 9;
x = 3;
12 * 3 = 36 centimeters – length of the AC leg;
5 * 3 = 15 centimeters – the length of the BC leg.
Answer: 36 centimeters; 15 centimeters.