2024 AMC 8 Problems/Problem 24

Problem

Jean has 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 while the other peak is $12$ feet high. Each peak forms a $90^\circ$ angle, and the straight sides form a $45^\circ$ angle with the ground. The artwork has an area of $183$ square feet. The sides of the mountain meet at an intersection point near the center of the artwork, $h$ feet above the ground. What is the value of $h?$

Solution 1

Extend the "inner part" of the mountain so that the image is two right triangles that overlap in a third right triangle. The side length of the largest right triangle is $12\sqrt{2},$ which means its area is $144.$ Similarly, the area of the second largest right triangle is $64$ (the side length is $8\sqrt{2}$ ), and the area of the overlap triangle is $h^2$ (the side length is $h\sqrt{2}$ ) Thus, $144+64-h^2=183,$ which means that $\boxed{h=5}.$

~BS2012

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2024 AMC 8 Problems/Problem 24

Problem

Solution 1