The base ab of an isosceles triangle abc is 18 cm and the lateral side bc is 15 cm.

The base ab of an isosceles triangle abc is 18 cm and the lateral side bc is 15 cm. Find the radii of the circles inscribed in the triangle and circumscribed around the triangle, the answers should be: r = 4.5 cm R = 9.375 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.
p = (a + b + b) / 2 = (a + 2b) / 2,
where a is the base of an isosceles triangle, b is the lateral side of an isosceles triangle.
p = (18 + 2 * 15) / 2 = 48/2 = 24.
The area of ​​an isosceles triangle according to Heron’s formula:
S = (p – b) * √p (p – a);
S = (24 – 15) * √24 (24 – 18) = 9√24 * 6 = 9√144 = 9 * 12 = 108 (cm ^ 2).
Inscribed circle radius:
r = 108/24 = 4.5 (cm).
2. The radius of a circle circumscribed about a triangle is found by the formula:
R = abc / 4S = (a * b ^ 2) / 4S.
Radius of the circumscribed circle:
R = (18 * 15 ^ 2) / 4 * 108 = 18 * 225/4 * 108 = 4050/432 = 9.375 (cm).
Answer: r = 4.5 cm, R = 9.375 cm.