2007 AMC 12A Problem 10

Problem 10 of 25EasierGeometry

A triangle with side lengths in the ratio

3:4:5

is inscribed in a circle of radius

3.

What is the area of the triangle?

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Solution

Let the sides be

3x,

4x,

and

5x.

The triangle is right, so its hypotenuse is a diameter. Thus

5x=2\cdot 3=6,

giving

x=\tfrac65.

The area is

\tfrac12\cdot 3x\cdot 4x=6x^2

=6\cdot\tfrac{36}{25}

=\tfrac{216}{25}=8.64.

Thus, the correct answer is A.

Concepts: right triangle · circumcircle, circumcenter, and circumradius · triangle area

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.