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: Calculate \( b \) and \( \theta \).
#### Given Triangle:
- Right-angled at one of its vertices.
- One leg of the right triangle is labeled as \( 8 \) units.
- The hypotenuse of the triangle is labeled as \( 22 \) units.
- The other leg of the triangle is labeled as \( b \).
### Steps to Solve:
1. **Using Pythagorean Theorem to find \( b \):**
The Pythagorean Theorem states:
\[
a^2 + b^2 = c^2
\]
where \( a \) and \( b \) are the legs of the right triangle and \( c \) is the hypotenuse.
In this case:
\[
8^2 + b^2 = 22^2
\]
Solving for \( b \):
\[
64 + b^2 = 484
\]
\[
b^2 = 420
\]
\[
b = \sqrt{420}
\]
\[
b \approx 20.49 \quad \text{(rounded to two decimal places)}
\]
2. **Finding the angle \( \theta \):**
To find \( \theta \), we can use trigonometric ratios. Here, we use the sine ratio:
\[
\sin(\theta) = \frac{\text{opposite leg}}{\text{hypotenuse}} = \frac{8}{22}
\]
Solving for \( \theta \):
\[
\theta = \sin^{-1}\left(\frac{8}{22}\right)
\]
\[
\theta \approx \sin^{-1}(0.3636)
\]
\[
\theta \approx 21.36^\circ \quad \text{(rounded to two decimal places)}
\]
### Conclusion:
- \( b \approx 20.49 \)
- \( \theta \approx 21.36^\circ \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F991e73b4-6c7b-4bec-a3ba-61e369206183%2Fd4f70783-f657-42a7-be78-e6fd4fb85287%2F1chyg54_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem: Calculate \( b \) and \( \theta \).
#### Given Triangle:
- Right-angled at one of its vertices.
- One leg of the right triangle is labeled as \( 8 \) units.
- The hypotenuse of the triangle is labeled as \( 22 \) units.
- The other leg of the triangle is labeled as \( b \).
### Steps to Solve:
1. **Using Pythagorean Theorem to find \( b \):**
The Pythagorean Theorem states:
\[
a^2 + b^2 = c^2
\]
where \( a \) and \( b \) are the legs of the right triangle and \( c \) is the hypotenuse.
In this case:
\[
8^2 + b^2 = 22^2
\]
Solving for \( b \):
\[
64 + b^2 = 484
\]
\[
b^2 = 420
\]
\[
b = \sqrt{420}
\]
\[
b \approx 20.49 \quad \text{(rounded to two decimal places)}
\]
2. **Finding the angle \( \theta \):**
To find \( \theta \), we can use trigonometric ratios. Here, we use the sine ratio:
\[
\sin(\theta) = \frac{\text{opposite leg}}{\text{hypotenuse}} = \frac{8}{22}
\]
Solving for \( \theta \):
\[
\theta = \sin^{-1}\left(\frac{8}{22}\right)
\]
\[
\theta \approx \sin^{-1}(0.3636)
\]
\[
\theta \approx 21.36^\circ \quad \text{(rounded to two decimal places)}
\]
### Conclusion:
- \( b \approx 20.49 \)
- \( \theta \approx 21.36^\circ \)
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