In triangle ABC, angle C is 90, AC = 36, BC = 10.5. Find the radius of the inscribed circle.

If a triangle has a right angle (angle C is 90 degrees), then it is a right-angled triangle. The radius of the inscribed circle is calculated by the formula r = √ (((p – a) (p – b) (p – c)) / p), where p = (a + b + c) / 2; a, b, c – sides of the triangle.

AC and BC are legs, AB is hypotenuse; find the hypotenuse by the Pythagorean theorem: The square of the hypotenuse is equal to the sum of the squares of the legs;

AB ^ 2 = AC ^ 2 + BC ^ 2;

AB ^ 2 = 36 ^ 2 + 10.5 ^ 2 = 1296 + 110.25 = 1406.25;

AB = 37.5.

p = (36 + 10.5 + 37.5) / 2 = 84/2 = 42;

r = √ (((42 – 36) (42 – 10.5) (42 – 37.5)) / 42) = √ ((6 * 31.5 * 4.5) / 42) = √ (850, 5/42) = √20.25 = 4.5.

Answer. 4.5.