Question
5. The length of one leg of a right triangle is 5 centimeters shorter than the hypotenuse. The hypotenuse is 15 centimeters. What is the length of the unknown leg of the right triangle rounded to the nearest tenth?
Ask by Gibson Johnston. in the United States
Mar 12,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The unknown leg of the right triangle is approximately 11.2 centimeters.
Solution
Given that the hypotenuse has length \(15\) centimeters and one leg is \(5\) centimeters shorter than the hypotenuse, the known leg is
\[
15 - 5 = 10 \text{ cm}.
\]
Let the unknown leg be \(x\) centimeters. By the Pythagorean theorem we have
\[
x^2 + 10^2 = 15^2.
\]
Substitute the known values:
\[
x^2 + 100 = 225.
\]
Subtract \(100\) from both sides:
\[
x^2 = 225 - 100 = 125.
\]
Taking the square root of both sides gives:
\[
x = \sqrt{125} = 5\sqrt{5}.
\]
Approximating using \(\sqrt{5} \approx 2.236\):
\[
x \approx 5 \times 2.236 = 11.18 \text{ cm}.
\]
Rounded to the nearest tenth, the length of the unknown leg is
\[
11.2 \text{ cm}.
\]
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Simplify this solution Extra Insights
In this right triangle, we know that the hypotenuse measures 15 centimeters. According to the problem, one leg is 5 centimeters shorter than the hypotenuse, so we can find that leg's length by calculating 15 cm - 5 cm, which equals 10 cm. Now, we can let the unknown leg be represented as \( x \). Using the Pythagorean theorem: \[ x^2 + 10^2 = 15^2 \] This simplifies to: \[ x^2 + 100 = 225 \] Now, subtract 100 from both sides: \[ x^2 = 125 \] Taking the square root gives: \[ x = \sqrt{125} \approx 11.2 \text{ centimeters} \] So, the length of the unknown leg of the right triangle is approximately 11.2 centimeters.
