jmc

Question

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^ angle, and the straight sides form a 45^ 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? FIGURE

Accepted Answer

Extend the "inner part" of the mountain so that the image is two right triangles that overlap in a third right triangle as shown. FIGURE The side length of the largest right triangle is 122, which means its area is 144. Similarly, the area of the second largest right triangle is 64 (the side length is 82), and the area of the overlap is h^2 (the side length is h2). Thus, 144+64-h^2=183, which means that the answer is 5.