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

The base of an isosceles triangle is 18 cm and the lateral side is 15 cm. Find the radii of the inscribed and circumscribed circles around the triangle.

In an isosceles triangle, a is the sides, b is the base, h is the height lowered to the base, r is the radius of the inscribed circle, R is the radius of the circumscribed circle.
1. Find the radius of the inscribed circle:
r = b / 2 * √ ((2a – b) / (2a + b));
r = 18/2 * √ ((2 * 15 – 18) / (2 * 15 + 18));
r = 9 * √ ((30 – 18) / (30 + 18);
r = 9 * √ (12/48);
r = 9 * √ (1/4);
r = 9/2;
r = 4.5 cm.
2. The radius of the circumscribed circle is found by the formula:
R = a ^ 2 / 2h
Find h. h divides the given triangle into two equal right-angled triangles with one of the legs equal to half the base of the original triangle (since the height in an isosceles triangle is also the median). Then the height is found by the Pythagorean theorem:
h = √ (a ^ 2 – (b / 2) ^ 2);
h = √ (15 ^ 2 – (18/2) ^ 2);
h = √ (225 – 9 ^ 2);
h = √ (225 – 81);
h = √144;
h = 12 cm.
Find the radius of the circumscribed circle:
R = 15 ^ 2/2 * 12 = 225/24 = 9.375 (cm).
Answer: r = 4.5 cm, R = 9, 375 cm.