2022 AMC 12B Problem 13

Question

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?

15\dfrac18

15\dfrac38

15\dfrac12

15\dfrac58

15\dfrac78

Answer: D

Concepts:coordinate geometry area decomposition Pythagorean Triple

Difficulty rating: 1660

Small hint:

Set up coordinates; the tilt of the square makes

3

-

4

-

5

right triangles at the corners

Big hint:

The whole square lies inside the rectangle except one triangle sticking out past the top edge; subtract that triangle from

25

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.