Find the legs of a right-angled triangle if its bisector divides the hypotenuse into parts equal to 12 cm and 5 cm.

Find the length of the hypotenuse of the triangle:
12 + 5 = 17 cm.

We denote the legs of the triangle by x and y.
We use the theorem on the bisector of the inner angle and compose the first equation:
x / y = 5/12;
y = 12 * x / 5;
y = 2.4 * x.

Using the Pythagorean theorem, we get the second equation:
x² + y² = 17².

Substitute the value of y from the first equation into the second:
x² + (2.4 * x) ² = 289;
x² + 5.76 * x² = 289;
x² = 289 / 6.76;
x = 85/13 = 6 7/13 cm – the first leg of the triangle.

Find the second leg:
y = 2.4 * 85/13 = 204/13 = 15 9/13 cm.

Answer: the legs of the triangle are 6 7/13 cm and 15 9/13 cm.