Solution of the Week #503 - Semicircle in a Triangle

If we split the triangle into two triangles by drawing a line from the apex to the centre of the semicircle, we can see that the ‘height’ of each triangle is equal to the radius. So the total area of the triangle is 87r/2 + 75r/2 = 81r

 We can also calculate the area of the triangle using Heron’s formula, which states that the area is equal to the square root of the product s(s-a)(s-b)(s-c), where a, b and c are the sides of the triangle and s is the semi-perimeter.

 s=(87+75+108)/2 = 135

s(s-a)(s-b)(s-c) = 135*48*60*27 = 10497600

Area = sqrt(10497600) = 3240

 Now we have two different expressions for the triangle’s area we can equate them:

81r = 3240

And then find the radius

r = 3240/81 = 40.