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2007 AMC 10A Problem 14

Problem 14 of 25IntermediateGeometry

A triangle with side lengths in the ratio 3:4:53 : 4 : 5 is inscribed in a circle of radius 3.3. What is the area of the triangle?

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Solution

Let the sides be 3x,4x,5x.3x, 4x, 5x. The triangle is right-angled, so its hypotenuse is a diameter. Thus 5x=2⋅3=6,5x = 2 \cdot 3 = 6, giving x=65.x = \tfrac65. The area is 12(3x)(4x)=6x2=6⋅3625=8.64. \begin{aligned} \tfrac12 (3x)(4x) &= 6x^2 = 6 \cdot \tfrac{36}{25} \\ &= 8.64. \end{aligned} Thus, the correct answer is A.

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Concepts: inscribed angle · right triangle · triangle area

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