Let's take a look at the given diagram. We are given that AB=AC=12 and BC=8, so the drawn triangle is isosceles. Additionally we know that all three segments are tangent to ∘ P.
First let's evaluate the height of the triangle h. To do this let's recall that the height in an isosceles triangle bisects the base.
Let's write an equation for h using the Pythagorean Theorem. Notice that since h represents the height, we will consider only positive case when taking a square root of h^2.
The height of the triangle is approximately 11.3 units. Using this value, we can evaluate the area of a triangle.
A_(△ ABC)=1/2(11.3)(8)=45.2
The area of the triangle is 45.2 square units. Next let's recall that tangent segments are perpendicular to the radii at the tangency points. Let's call the radius of the circle r.
If we connect each vertex with the center of a triangle, then we have three triangles each with the height of r.
Notice that the sum of the areas of these three triangles needs to be equal to the area of △ ABC, 45.2. We will use the formula for the area of a triangle to write an equation and solve for r.
A_(△ ABC)=A_(△ BPC)+A_(△ BPA)+A_(△ APC) ⇓
45.2=1/2(8)r+1/2( 12)r+1/2( 12)r
Let's solve the equation to find the radius.
