What is the diameter of the inscribe circle of the triangle?

Work Shown:

The two angle bisectors intersect to form the incenter. The incenter is the center of the inscribed circle (aka "incircle"). This circle is the largest possible, but is completely inside the triangle. No portions of the circle spill outside the triangle.

The two segments marked with lengths x+8 and 2x-5 are two radii of the incircle. This is because the sides of the triangle are tangent to the incircle.

Since all radii are the same length, we can set those expressions equal to one another and solve for x

x+8 = 2x-5

x-2x = -5-8

-x = -13

x = 13

This x value then leads to

Both radii are 21 units long, which helps confirm we have the proper x value. We double the length of the radius ot get the diameter.

So the diameter of the incircle is 2*21 = 42 units long.

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step-by-step explanation:

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