The sides of the triangle are 10 10 12. Find the radius of the inscribed circle.

Given: triangle, a = b = 10, c = 12.

Find: r – radius of the inscribed circle in the given triangle.

Solution: a = b, therefore, the triangle is isosceles.

Additional construction: we drop the height h to the side c, h – the median – divides the base c into two equal parts; from a right-angled triangle with sides 6 and 10 we find h, h = √ (100 – 36) = 8;

S = 8 * 12: 2 = 48 (square units), the area of ​​this triangle.

r = (2 * S): P, where P is the perimeter of the triangle, P = 10 + 10 + 12 = 32 (units),

r = (2 * 48): 32 = 3 (units).

Knowing the three sides of the triangle, the area can be calculated using Heron’s formula, and since the calculation requires a half-perimeter, the radius can be immediately found using the formula r = S: p, where p is the half-perimeter of the triangle.
Answer: the radius of the circle inscribed in the triangle is 3.