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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![**Problem:**
Solve for \( x \). Round to the nearest tenth, if necessary.
**Diagram Explanation:**
The image contains a right-angled triangle labeled \( \triangle KLM \). The right angle is at vertex \( L \). The side \( KL \) is adjacent to the given angle, \( \angle KML \), which measures 27 degrees.
- **Angle \( \angle KML \) = 27°**
- **Side \( KL \) = 6.3 units**
- **Hypotenuse \( KM \) = \( x \) units (unknown to be solved for)**
**Solution:**
To find the length of the hypotenuse \( x \), we use the cosine trigonometric ratio, since we have the adjacent side and need to find the hypotenuse.
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
Substitute the values:
\[
\cos(27^\circ) = \frac{6.3}{x}
\]
This can be rearranged to solve for \( x \):
\[
x = \frac{6.3}{\cos(27^\circ)}
\]
Using a calculator to find the cosine of 27 degrees:
\[
\cos(27^\circ) \approx 0.8910
\]
Thus,
\[
x = \frac{6.3}{0.8910} \approx 7.1
\]
So, the length of \( x \), rounded to the nearest tenth, is approximately \( 7.1 \text{ units} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F16ab444c-cc67-4b2e-a5c5-3a02b2df29f8%2Fd392b4e6-76bb-4757-b6ad-46798ed813dd%2Fmfu6kt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem:**
Solve for \( x \). Round to the nearest tenth, if necessary.
**Diagram Explanation:**
The image contains a right-angled triangle labeled \( \triangle KLM \). The right angle is at vertex \( L \). The side \( KL \) is adjacent to the given angle, \( \angle KML \), which measures 27 degrees.
- **Angle \( \angle KML \) = 27°**
- **Side \( KL \) = 6.3 units**
- **Hypotenuse \( KM \) = \( x \) units (unknown to be solved for)**
**Solution:**
To find the length of the hypotenuse \( x \), we use the cosine trigonometric ratio, since we have the adjacent side and need to find the hypotenuse.
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
Substitute the values:
\[
\cos(27^\circ) = \frac{6.3}{x}
\]
This can be rearranged to solve for \( x \):
\[
x = \frac{6.3}{\cos(27^\circ)}
\]
Using a calculator to find the cosine of 27 degrees:
\[
\cos(27^\circ) \approx 0.8910
\]
Thus,
\[
x = \frac{6.3}{0.8910} \approx 7.1
\]
So, the length of \( x \), rounded to the nearest tenth, is approximately \( 7.1 \text{ units} \).
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