What is the radius of the circle inscribed in triangle ABC if AB=5, AC=6, BC=7? Express your answer in simplest radical form.

MIRB15
Jul 7, 2017

#1**+1 **

The radius of the largest circle that will fit inside the triangle is given by :

2A / P

Where A is the area of the triangle and P is the perimeter

To calculate the area we can use Heron's formula

s = the semi-perimeter = [ 5 + 6+ 7] / 2 = 18/2 = 9

And the area is given by

√ [ s (s - a) (s - b) (s - c) ] where a, b, c are the side lengths of the triangle

So we have

√ [ 9 (9 - 5) (9 - 6) (9 - 7) ] = √ [ 9 * 4 * 3 * 2 ] = √[36 * 6 ] = 6√6

And the perimeter is 18

So.....the radius is given by

2 [ 6√6] / 18 =

(2/3)√6

CPhill
Jul 7, 2017

#2**+1 **

Here's how the formula is derived :

Since the radius, r, of the incircle can be drawn to meet each side of the larger triangle at right angles, then we can construct three triangles by connecting each vertex of the larger triangle to its incenter. And each of these triangles has a height of r and a side as a base.

So.....the total area, A, of the the larger triangle is

(1/2) r ( a + b + c) = A where a,b,c are the side lengths

Multiply both sides by 2

r (a + b + c) = 2A

Divide both sides by (a + b + c)

r = 2A / (a + b + c)

But (a + b + c) is the perimeter = P

So....

r = 2A / P

CPhill
Jul 7, 2017