Question

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?

$X + Z = W + Y$

$W + X = Z$

$3X + 4Y = 5Z$

$X + W = \dfrac{1}{2}(Y + Z)$

$X + Y = Z$

Answer: E

Concepts:Pythagorean Theorem area

Difficulty rating: 1310

Solution:

For a right isosceles triangle built on a side of length $s$ , the two legs have length $s$ , so its area is $s^2/2$ .

$\begin{aligned} &W=\frac{3\cdot4}{2}=6, \\ &\quad X=\frac{3^2}{2}=4.5, \\ &\quad Y=\frac{4^2}{2}=8, \\ &\quad Z=\frac{5^2}{2}=12.5. \end{aligned}$

These values satisfy $X+Y=Z$ , and the other listed equations do not.

Thus, E is the correct answer.

2002 AMC 8 Problem 16

Answer: E

Difficulty rating: 1310

Solution:

Problem 16 in Other Years