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University of South Carolina High School Math Contest/1993 Exam/Problem 27

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Problem

Suppose $\triangle ABC$ is a triangle with area 24 and that there is a point $P$ inside $\triangle ABC$ which is distance 2 from each of the sides of $\triangle ABC$ . What is the perimeter of $\triangle ABC$ ?

$\mathrm{(A) \ } 12 \qquad \mathrm{(B) \ }24 \qquad \mathrm{(C) \ }36 \qquad \mathrm{(D) \ }12\sqrt{2} \qquad \mathrm{(E) \ }1...$

Solution

Notice that $P$ is the incenter of the triangle. The incircle has radius $2$ . Thus, using the area formula $A=rs$ we have $2 \cdot s=24 \Longrightarrow s=12$ and the perimeter is $24$ .

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