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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![### Finding Angle m∠X
In this problem, we are given a right triangle XYZ with a right angle ∠Y. The lengths of the sides adjacent to the right angle are given as follows:
- XY = 5 units
- YZ = 12 units
- XZ = 13 units
We need to find the measure of angle ∠X.
**Triangle Description:**
- The triangle includes three vertices labeled X, Y, and Z.
- The right angle is located at vertex Y.
- The length of side XY (adjacent to ∠X) is 5 units.
- The length of side YZ (opposite to ∠X) is 12 units.
- The length of the hypotenuse XZ is 13 units.
**Objective:**
Calculate the measure of angle ∠X (m∠X).
**Multiple Choice Options:**
1. 21.04°
2. 22.62°
3. 67.38°
4. 68.96°
**Solution:**
To find angle X, you can use trigonometric ratios. Specifically, you can use the sine function or the cosine function since you are given the lengths of the sides of the triangle.
Let's use the sine function for ∠X:
\[ \sin(X) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{YZ}{XZ} = \frac{12}{13} \]
To find X, take the inverse sine (arcsine) of 12/13:
\[ X = \sin^{-1} \left(\frac{12}{13}\right) \approx 67.38^\circ \]
So, the correct measure of angle ∠X is:
\[ \boxed{67.38^\circ} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5e777900-48b9-4d96-bed3-a41476812193%2F40df354b-1ead-47f1-afe3-ac1c035f0e75%2Fyng7kbw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding Angle m∠X
In this problem, we are given a right triangle XYZ with a right angle ∠Y. The lengths of the sides adjacent to the right angle are given as follows:
- XY = 5 units
- YZ = 12 units
- XZ = 13 units
We need to find the measure of angle ∠X.
**Triangle Description:**
- The triangle includes three vertices labeled X, Y, and Z.
- The right angle is located at vertex Y.
- The length of side XY (adjacent to ∠X) is 5 units.
- The length of side YZ (opposite to ∠X) is 12 units.
- The length of the hypotenuse XZ is 13 units.
**Objective:**
Calculate the measure of angle ∠X (m∠X).
**Multiple Choice Options:**
1. 21.04°
2. 22.62°
3. 67.38°
4. 68.96°
**Solution:**
To find angle X, you can use trigonometric ratios. Specifically, you can use the sine function or the cosine function since you are given the lengths of the sides of the triangle.
Let's use the sine function for ∠X:
\[ \sin(X) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{YZ}{XZ} = \frac{12}{13} \]
To find X, take the inverse sine (arcsine) of 12/13:
\[ X = \sin^{-1} \left(\frac{12}{13}\right) \approx 67.38^\circ \]
So, the correct measure of angle ∠X is:
\[ \boxed{67.38^\circ} \]
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