Question

Circle O is inscribed in the given triangle.

Circle O is inscribed within a triangle. Points Q, P, and R of the circle are on the sides of the triangle. Point P splits the side of the triangle into lengths of 12 and 4. Point Q splits the sides of the triangle into lengths of blank and 6.

What is the perimeter of the triangle?

22 units
30 units
44 units
60 units

Answer 1

Let the triangle sides be a, b, c and s the semiperimeter. The incircle tangency splits sides into lengths s−a, s−b, s−c.

A side split into 12 and 4 means two of {s−a, s−b, s−c} are 12 and 4. If s−b = 12 and s−c = 4 then that side length a = (s−b)+(s−c) = 16. The remaining value is s−a.

The other tangency gives one segment 6, so s−a = 6 ⇒ s = 22. Thus the perimeter is 2s = 44.

Answer: 44 units.