2008 AMC 10A Problem 18

Problem 18 of 25IntermediateAlgebra Geometry

A right triangle has perimeter

32

and area

20.

What is the length of its hypotenuse?

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Solution

Let the legs be

y, z

and the hypotenuse

x.

Then

y^2 + z^2 = x^2,

y + z = 32 - x,

and

yz = 40.

Squaring the second equation,

\begin{aligned} &(32 - x)^2 \\ &\quad = y^2 + z^2 + 2yz = x^2 + 80. \end{aligned}

This gives

1024 - 64x = 80,

x = \dfrac{59}{4}.

Thus, the correct answer is B.

Concepts: Pythagorean Theorem · system of equations · algebraic manipulation

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