Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 27"
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== Problem == | == Problem == | ||
| + | Suppose <math>\triangle ABC</math> is a triangle with area 24 and that there is a point <math>P</math> inside <math>\triangle ABC</math> which is distance 2 from each of the sides of <math>\triangle ABC</math>. What is the perimeter of <math>\triangle ABC</math>? | ||
| − | <center><math> \mathrm{(A) \ } \qquad \mathrm{(B) \ } \qquad \mathrm{(C) \ } \qquad \mathrm{(D) \ } \qquad \mathrm{(E) \ } </math></center> | + | <center><math> |
| + | \mathrm{(A) \ } 12 \qquad \mathrm{(B) \ }24 \qquad \mathrm{(C) \ }36 \qquad \mathrm{(D) \ }12\sqrt{2} \qquad \mathrm{(E) \ }12\sqrt{3} </math></center> | ||
== Solution == | == Solution == | ||
| + | Notice that <math>P</math> is the [[incenter]] of the [[triangle]]. The [[incircle]] has [[radius]] <math>2</math>. Thus, using <math>rs=A</math>, we have <math>2 \cdot s=24 \Longrightarrow s=12</math> and the perimeter is <math>24</math>. | ||
== See also == | == See also == | ||
* [[University of South Carolina High School Math Contest/1993 Exam]] | * [[University of South Carolina High School Math Contest/1993 Exam]] | ||
| + | |||
| + | [[Category:Intermediate Geometry Problems]] | ||
is a triangle with area 24 and that there is a point 
inside 
. Thus, using 
, we have 
and the perimeter is 
.
