2002 AMC 8 Problems/Problem 16

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Problem

Right isosceles triangles are constructed on the sides of a 3-4-5 right triangle, as shown. A capital letter represents the area of each triangle. Which one of the following is true?

[asy] /* AMC8 2002 #16 Problem */ draw((0,0)--(4,0)--(4,3)--cycle); draw((4,3)--(-4,4)--(0,0)); draw((-0.15,0.1)--(0,0.25)--(.15,0.1)); draw((0,0)--(4,-4)--(4,0)); draw((4,0.2)--(3.8,0.2)--(3.8,-0.2)--(4,-0.2)); draw((4,0)--(7,3)--(4,3)); draw((4,2.8)--(4.2,2.8)--(4.2,3)); label(scale(0.8)*"$Z$", (0, 3), S); label(scale(0.8)*"$Y$", (3,-2)); label(scale(0.8)*"$X$", (5.5, 2.5)); label(scale(0.8)*"$W$", (2.6,1)); label(scale(0.65)*"5", (2,2)); label(scale(0.65)*"4", (2.3,-0.4)); label(scale(0.65)*"3", (4.3,1.5));[/asy]

$\textbf{(A)}\ X+Z=W+Y\qquad\textbf{(B)}\ W+X=Z\qquad\textbf{(C)}\ 3X+4Y=5Z\qquad$ $\textbf{(D)}\ X+W=\frac{1}{2}(Y+Z)\qquad\textbf{(E)}\ X+Y=Z$

Solution

The area of a right triangle can be found by using the legs of triangle as the base and height. In the three isosceles triangles, the length of their second leg is the same as the side that is connected to the $3-4-5$ triangle.

\begin{align*} W&=(3)(4)/2 = 6\\ X&=(3)(3)/2=4.5\\ Y&=(4)(4)/2 = 8\\ Z&=(5)(5)/2 = 12.5 \end{align*}

Plugging into the answer choices, the only that works is $\boxed{\textbf{(E)}\ X+Y=Z}$.

See Also

2002 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 15
Followed by
Problem 17
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All AJHSME/AMC 8 Problems and Solutions
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