2022 AMC 12B Problem 13

Problem 13 of 25IntermediateGeometry

The diagram below shows a rectangle with side lengths

4

and

8

and a square with side length

5.

Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?

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Solution

Place the rectangle as

[0,8] \times [0,4].

The tilted square, using the

3

4

5

right triangles, has vertices

(4, 0),

(0, 3),

(3, 7),

and

(7, 4).

The entire square lies inside the rectangle except for the triangle poking above the top edge

y = 4.

That triangle has vertices

(0.75, 4),

(3, 7),

and

(7, 4),

with area

\dfrac12 \cdot (7 - 0.75) \cdot (7 - 4) = \dfrac{75}{8}.

The region inside both is

25 - \dfrac{75}{8} = \dfrac{125}{8} = 15\dfrac58.

Thus, the correct answer is D.

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Tagged: coordinate geometry · area decomposition · Pythagorean Triple

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