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
Related questions
Question
![**Title: Solving for an Angle in a Right Triangle**
**Objective:** Learn how to determine the measure of an angle in a right-angled triangle using the given lengths of the hypotenuse and one adjacent side.
### Problem Statement
**Solve for \( x \). Round to the nearest tenth of a degree, if necessary.**
### Given Diagram:
A right-angled triangle \( \triangle KLM \) is presented with the following details:
- Segment \( KM \) is the hypotenuse, labeled with a length of \( 6.5 \) units.
- Segment \( LM \) is one of the legs (adjacent to angle \( x^\circ \)), labeled with a length of \( 4.4 \) units.
- The right angle is located at vertex \( L \).
### Solution Approach
To find the angle \( x \):
1. **Trigonometric Ratio**: Use the cosine function, which relates the adjacent side and the hypotenuse in a right-angled triangle.
\[
\cos(x) = \frac{\text{adjacent side}}{\text{hypotenuse}}
\]
For \( \triangle KLM \):
\[
\cos(x) = \frac{LM}{KM} = \frac{4.4}{6.5}
\]
2. **Calculate the Cosine Value**:
\[
\cos(x) = \frac{4.4}{6.5} \approx 0.6769
\]
3. **Determine the Angle**: Use the inverse cosine function (often denoted as \( \cos^{-1} \) or \( \text{arccos} \)) to find the angle:
\[
x = \cos^{-1}(0.6769)
\]
4. **Use a Calculator to Find \( x \)**:
\[
x \approx 47.1^\circ
\]
### Conclusion
The value of \( x \) is approximately \( 47.1 \) degrees when rounded to the nearest tenth of a degree.
**Note:** Ensure the calculator is set to degree mode to get the correct measure of the angle in degrees.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb223a19b-0fd0-443c-881a-6f762159fad5%2F9f81335f-cb4b-4bcd-8f54-b0f7e92ab1fc%2Fiphsug_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Solving for an Angle in a Right Triangle**
**Objective:** Learn how to determine the measure of an angle in a right-angled triangle using the given lengths of the hypotenuse and one adjacent side.
### Problem Statement
**Solve for \( x \). Round to the nearest tenth of a degree, if necessary.**
### Given Diagram:
A right-angled triangle \( \triangle KLM \) is presented with the following details:
- Segment \( KM \) is the hypotenuse, labeled with a length of \( 6.5 \) units.
- Segment \( LM \) is one of the legs (adjacent to angle \( x^\circ \)), labeled with a length of \( 4.4 \) units.
- The right angle is located at vertex \( L \).
### Solution Approach
To find the angle \( x \):
1. **Trigonometric Ratio**: Use the cosine function, which relates the adjacent side and the hypotenuse in a right-angled triangle.
\[
\cos(x) = \frac{\text{adjacent side}}{\text{hypotenuse}}
\]
For \( \triangle KLM \):
\[
\cos(x) = \frac{LM}{KM} = \frac{4.4}{6.5}
\]
2. **Calculate the Cosine Value**:
\[
\cos(x) = \frac{4.4}{6.5} \approx 0.6769
\]
3. **Determine the Angle**: Use the inverse cosine function (often denoted as \( \cos^{-1} \) or \( \text{arccos} \)) to find the angle:
\[
x = \cos^{-1}(0.6769)
\]
4. **Use a Calculator to Find \( x \)**:
\[
x \approx 47.1^\circ
\]
### Conclusion
The value of \( x \) is approximately \( 47.1 \) degrees when rounded to the nearest tenth of a degree.
**Note:** Ensure the calculator is set to degree mode to get the correct measure of the angle in degrees.
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