SOLUTION: What is the radius of the largest circle that will fit inside a triangle that has two 15-inch sides and an 18-inch side?

The given triangle is isosceles with congruent lateral sides of 15 inches long
and the base of 18 inches.


The altitude of this triangle drawn to the 18-inches side, divide this triangle in two right-angled triangles.


This altitude is one leg of these triangles, while the other legs are 18/2 = 9 inches long.


From the Pythagorean theorem, the length of the altitude is   = 12 inches.


Then the area of the given triangle is   = 9*12 = 108 sq.inches.


Let the radius of the inscribed circle be r.


Now use the formula  

    area = 

for the triangle, where P is the perimeter P of the triangle   P = 15+15+12 = 42 inches.


The area of the triangle is 108 sq.inches.


Hence,  108 = ,  which gives

    r =  =  =  =  inches.   ANSWER