Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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Question
![**Problem Statement:**
Given a right-angled triangle XYZ, where XY = 10 inches, XZ = 12 inches, and angle X is the right angle, determine the best approximation for the measure of angle XYZ.
**Choices:**
- 33.6°
- 39.8°
- 50.2°
- 56.4°
**Explanation:**
We are given a right triangle XYZ. In this triangle:
- XY is one of the legs and is 10 inches long.
- XZ is the hypotenuse and is 12 inches long.
- YZ is the other leg, denoted as 'x' inches.
Our goal is to find the best approximation for angle XYZ.
To find angle XYZ, we can use the trigonometric relationship involving the lengths of the sides. Specifically, the cosine of angle XYZ is given by the adjacent side (XY) over the hypotenuse (XZ).
\[ \cos (\angle XYZ) = \frac{ \text{adjacent} }{ \text{hypotenuse} } = \frac{XY}{XZ} = \frac{10}{12} \]
Simplifying,
\[ \cos (\angle XYZ) = \frac{5}{6} \]
To find the angle whose cosine is \(\frac{5}{6}\), we use the inverse cosine function:
\[ \angle XYZ = \cos^{-1}\left(\frac{5}{6}\right) \]
Using a calculator, we find:
\[ \angle XYZ \approx 33.6° \]
Therefore, the best approximation for the measure of angle XYZ is:
**33.6°**
**Graph/Diagram Explanation:**
The diagram on the left side of the problem statement shows a right-angled triangle with angle X being the right angle (denoted by the small square at angle X). The legs of the triangle are labeled with their respective lengths (10 inches and 12 inches), and the angle of interest, XYZ, is opposite to the 12-inch side. The possible measurements for this angle are listed as multiple-choice answers on the right side of the problem statement.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fefc4e4-0fc1-4d3a-92e0-1dd9cbc9acda%2F78cb7f11-6d63-48f9-9382-e72f6afdf798%2F1ww8in8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Given a right-angled triangle XYZ, where XY = 10 inches, XZ = 12 inches, and angle X is the right angle, determine the best approximation for the measure of angle XYZ.
**Choices:**
- 33.6°
- 39.8°
- 50.2°
- 56.4°
**Explanation:**
We are given a right triangle XYZ. In this triangle:
- XY is one of the legs and is 10 inches long.
- XZ is the hypotenuse and is 12 inches long.
- YZ is the other leg, denoted as 'x' inches.
Our goal is to find the best approximation for angle XYZ.
To find angle XYZ, we can use the trigonometric relationship involving the lengths of the sides. Specifically, the cosine of angle XYZ is given by the adjacent side (XY) over the hypotenuse (XZ).
\[ \cos (\angle XYZ) = \frac{ \text{adjacent} }{ \text{hypotenuse} } = \frac{XY}{XZ} = \frac{10}{12} \]
Simplifying,
\[ \cos (\angle XYZ) = \frac{5}{6} \]
To find the angle whose cosine is \(\frac{5}{6}\), we use the inverse cosine function:
\[ \angle XYZ = \cos^{-1}\left(\frac{5}{6}\right) \]
Using a calculator, we find:
\[ \angle XYZ \approx 33.6° \]
Therefore, the best approximation for the measure of angle XYZ is:
**33.6°**
**Graph/Diagram Explanation:**
The diagram on the left side of the problem statement shows a right-angled triangle with angle X being the right angle (denoted by the small square at angle X). The legs of the triangle are labeled with their respective lengths (10 inches and 12 inches), and the angle of interest, XYZ, is opposite to the 12-inch side. The possible measurements for this angle are listed as multiple-choice answers on the right side of the problem statement.
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