Find the radius of the inscribed circle and the radius of the circumscribed

Find the radius of the inscribed circle and the radius of the circumscribed circle for an isosceles triangle with a base of 10 cm and a side of 12 cm.

1. The radius of a circle inscribed in a triangle is found by the formula:
r = S / p,
where S is the area of the triangle, p is the semiperimeter of the triangle.
Semi-perimeter:
p = (a + 2b) / 2;
p = (10 + 2 * 12) / 2 = (10 + 24) / 2 = 34/2 = 17.
We find the area by Heron’s formula:
S = (p – b) * √p (p – a);
S = (17 – 12) * √17 (17 – 10) = 5√17 * 7 = 5√119 (cm square).
Inscribed circle radius:
r = 5√119 / 17 (cm).
2. The radius of a circle circumscribed about a triangle is found by the formula:
R = (a * b ^ 2) / 4S;
R = (10 * 12 ^ 2) / 4 * 5√119 = 1440 / 20√119 = 1440√119 / 20 * 119 = 1440√119 / 2380 = 72√119 / 119 (cm).
Answer: r = 5√119 / 17 cm, R = 72√119 / 119 cm.