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 $h?$

$4$

$5$

$4\sqrt{2}$

$6$

$5\sqrt{2}$

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Written solution:

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.

2024 AMC 8 Problem 24

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Problem 24 in Other Years