Triangles Unit Test part 1 A length of rope is stretched between the top edge of a building and a stake in the ground. The head of the stake is at ground level. The rope also touches a tree that is growing halfway between the stake and the building. If the tree is 16 feet tall, how tall is the building?
The Correct Answer and Explanation is:
To solve this problem, let’s break it down step by step using a geometric approach. We’ll assume the situation forms a right triangle, where:
- The building represents one vertical side of the triangle.
- The distance from the stake to the building forms the horizontal side.
- The rope stretched from the top of the building to the stake, touching the tree halfway, forms the hypotenuse.
Step 1: Understand the setup
- The rope touches the tree halfway between the stake and the building, so the horizontal distance from the stake to the tree is half the horizontal distance from the stake to the building.
- The tree is 16 feet tall, and we are asked to find the height of the building.
Step 2: Consider the geometric shape
The rope creates two similar triangles:
- One triangle is formed by the top of the building, the stake, and the point where the rope touches the tree.
- The second triangle is formed by the top of the building, the stake, and the point where the rope reaches the ground at the stake.
Both triangles are similar because they share the angle formed by the rope and the line from the stake to the building, and they both have a right angle.
Step 3: Use the properties of similar triangles
Since the triangles are similar, the ratios of corresponding sides are equal. Let’s denote:
- hhh as the height of the building.
- The horizontal distance between the stake and the building as ddd.
For the triangle involving the tree:
- The height of the tree is 16 feet.
- The horizontal distance from the stake to the tree is d2\frac{d}{2}2d.
We can write the proportion of corresponding sides from the similar triangles:hd=16d2\frac{h}{d} = \frac{16}{\frac{d}{2}}dh=2d16
Step 4: Solve the equation
Simplify the equation:hd=16d2=16×2d=32d\frac{h}{d} = \frac{16}{\frac{d}{2}} = \frac{16 \times 2}{d} = \frac{32}{d}dh=2d16=d16×2=d32
Now, multiply both sides by ddd to eliminate the denominator:h=32h = 32h=32
Final Answer:
The height of the building is 32 feet.
Explanation:
We used the concept of similar triangles, where the corresponding sides of the two triangles are proportional. By setting up the ratio of the height of the building to the total horizontal distance, and equating it to the ratio of the tree’s height to half the horizontal distance, we solved for the height of the building, finding that it is 32 feet tall.
