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Question

An altitude labeled x from the right angle to the hypotenuse divides a right triangle into two smaller right triangles. The altitude divides the hypotenuse into two segments labeled 10 inches and 15 inches.

Sarah is building a portable rabbit hutch in the shape of a triangular prism. A vertical support beam splits the hypotenuse of the right triangle into two pieces, one measuring 10 inches and one measuring 15 inches.

If the hutch is placed so that the hypotenuse of the triangle is on the ground, how tall is the hutch? Round your answer to the nearest tenth.

(1 point)
feet

Answer 1

To find the height \( x \) of the hutch, we can use the geometric mean relationship that arises when an altitude is drawn to the hypotenuse of a right triangle. The geometric mean states that:

\[ x^2 = a \cdot b \]

where \( x \) is the length of the altitude, \( a \) is one segment of the hypotenuse, and \( b \) is the other segment. In this scenario, the segments of the hypotenuse are given as \( a = 10 \) inches and \( b = 15 \) inches.

1. Calculate the product of the segments: \[ a \cdot b = 10 \cdot 15 = 150 \]

2. Now we can find \( x \): \[ x^2 = 150 \] \[ x = \sqrt{150} \]

3. Calculate \( \sqrt{150} \): \[ x \approx 12.247 \text{ inches} \]

To convert this to feet, note that: \[ 1 \text{ foot} = 12 \text{ inches} \] So: \[ x \text{ in feet} = \frac{12.247}{12} \approx 1.0206 \text{ feet} \]

Finally, rounding to the nearest tenth: \[ \text{Height of the hutch} \approx 1.0 \text{ feet} \]

Thus, the height of the hutch is approximately 1.0 feet.