Donna wants to measure the height of a tree. She sights the top of the tree, using a mirror that is lying flat on the ground. The mirror is 36ft from the tree, and Donna is standing 7.9ft from the mirror, as shown in the figure. Her eyes are 5ft above the ground. How tall is the tree? Round your answer to the nearest foot.
Question
Donna wants to measure the height of a tree. She sights the top of the tree, using a mirror that is lying flat on the ground. The mirror is 36ft from the tree, and Donna is standing 7.9ft from the mirror, as shown in the figure. Her eyes are 5ft above the ground. How tall is the tree? Round your answer to the nearest foot.
Solution
To solve this problem, we can use similar triangles. The two triangles are similar because the angle of incidence equals the angle of reflection, which means the two triangles share an angle, and they both have a right angle (90 degrees), so they are similar by AA (angle-angle).
The two triangles are:
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The small triangle formed by Donna, the mirror, and the line of sight to the top of the tree in the mirror. The height of this triangle is the height of Donna's eyes, 5ft, and the base is the distance from Donna to the mirror, 7.9ft.
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The large triangle formed by the tree, the mirror, and the line of sight from Donna's eyes to the top of the tree in the mirror. The base of this triangle is the total distance from Donna to the tree (the distance from Donna to the mirror plus the distance from the mirror to the tree), which is 7.9ft + 36ft = 43.9ft. The height of this triangle is the height of the tree, which we are trying to find.
Since the triangles are similar, the ratios of the corresponding sides are equal. So, we can set up the following equation to solve for the height of the tree (T):
5ft / 7.9ft = T / 43.9ft
Cross-multiplying gives us:
T = (5ft * 43.9ft) / 7.9ft
Solving this equation gives us the height of the tree, T.
T = 27.8ft
Rounding to the nearest foot, the tree is approximately 28 feet tall.
Similar Questions
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