2024 AMC 8 Problem 24

Question

Jean made a piece of stained glass art in the shape of two mountains, as shown in the figure below. One mountain peak is

8

feet high and the other peak is

12

feet high. Each peak forms a

90^\circ

angle, and the straight sides of the mountains form

45^\circ

angles with the ground. The artwork has an area of

183

square feet. The sides of the mountains meet at an intersection point near the center of the artwork,

h

feet above the ground. What is the value of

h?

4

5

4\sqrt{2}

6

5\sqrt{2}

Each mountain is built from a

45^\circ

-

45^\circ

-

90^\circ

triangle

Add the two large triangle areas and subtract the overlap

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Each mountain is a

45^\circ\text{-}45^\circ\text{-}90^\circ

triangle. A right isosceles triangle with height

x

has area

x^2

, since its two perpendicular sides each have length

x\sqrt2

.

The two large mountains have areas

8^2=64

and

12^2=144

. Their overlap is also a

45^\circ\text{-}45^\circ\text{-}90^\circ

triangle with height

h

, so its area is

h^2

. Thus

64+144-h^2=183,

so

h^2=25

, and

h=5

.

Thus, B is the correct answer.

Problem 24 in other years