# Geometry problem

• Oct 4th 2009, 01:49 PM
spred
Geometry problem
The perimeter of a triangle is 100. Find the numerical value of the area of the triangle divided by the radius of the inscribed circle of the triangle.
A. 100 B. 200 C. 25 D. 50 E. NOTA
• Oct 4th 2009, 03:07 PM
skeeter
Quote:

Originally Posted by spred
The perimeter of a triangle is 100. Find the numerical value of the area of the triangle divided by the radius of the inscribed circle of the triangle.
A. 100 B. 200 C. 25 D. 50 E. NOTA

let $\displaystyle a$, $\displaystyle b$, and $\displaystyle c$ be the lengths of the three sides of the triangle.

the perpendicular distance from each side to the incenter of the triangle equals the radius of the inscribed circle.

sketch a line segment from each vertex of the triangle to the incenter ... three triangles are formed inside the larger triangle, and each of the three areas of these smaller triangles sums to the area of the larger triangle.

$\displaystyle A = \frac{1}{2}ar + \frac{1}{2}br + \frac{1}{2}cr$

proceed from here ...