An isosceles triangle with base $24$ and legs of $15$ is inscribed in a circle. Find the radius.
- $7$
- $12\dfrac{1}{2}$
- $25$
- $25\dfrac{1}{2}$
-
cannot be determined
In an isosceles triangle with sides 15, 15, 24, the height to the base is sqrt(15^2 - 12^2) = sqrt(225 - 144) = sqrt(81) = 9. Using the circumradius formula R = (abc) / (4 * Area), Area = 0.5 * 24 * 9 = 108. R = (15 * 15 * 24) / (4 * 108) = 5400 / 432 = 12.5.
In this isosceles triangle, the altitude to the base is the square root of (15 squared minus 12 squared), which equals 9. Using the formula for the circumradius R equals the product of the sides divided by four times the area, we first find the area is half the base times the height, giving 0.5 times 24 times 9 for a total of 108. The radius is then the product of the sides (15 times 15 times 24) divided by (4 times 108), resulting in 5400 divided by 432. Simplifying this fraction gives a radius of 12 and a half.