Calculating radius of the circle

Question

A perfect circle sits exactly within a right-angled triangle,dividing its hypotenuse into two segments of 3 and 10 units.

The area of the triangle is 30 square units.

What's the radius of the circle, "r" ?

Answer 1

The center of the triangle is called the incenter. The circle is called the incircle. The center is the intersection of the angle bisectors.

using your drawing let A,B,C be the vertices on top, left and right respectively.Call the incenter I. let a,b,c be the points the circle intersects with sides BC,AC,AB respectively. Then Ab=Ac,Ba=Bc and Ca=Cb. Therefore AB=3+x and BC=10+x. Also $BC*AB=60$ so $(3+x)*(10+x)=60\rightarrow13x+x^2=30\rightarrow x=2$.

So now we know its a right triangle with sides $13,12,5$.

And know you can use Ye Olde formula $r=\sqrt{\frac{(s-a)(s-b)(s-c)}{s}} $

where s is half of the perimeter.