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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![### Understanding Right Triangles: Finding the Length of an Unknown Side
In this exercise, we are given a right triangle \( \triangle FDE \) with a right angle at vertex \( E \), an angle of \( 51^\circ \) at vertex \( D \), and the length of \( FE = 17 \) meters.
We need to find the length of side \( DE \).
#### Triangle Details:
- **Angle \( \angle EFD \)** (adjacent to side \( DE \)): \( 51^\circ \)
- **Side \( FE \)** (adjacent to the \( 51^\circ \) angle): \( 17 \) meters
- **Right angle** at \( E \): \( \angle DEF = 90^\circ \)
#### Solution:
To find side \( DE \), we can use trigonometric ratios. Specifically, we use the tangent ratio in right triangles, which relates the opposite side to the adjacent side.
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Here:
- \( \theta = 51^\circ \)
- Adjacent side \( FE = 17 \) meters
- Opposite side \( DE \) (which we need to find)
\[
\tan(51^\circ) = \frac{DE}{FE}
\]
\[
DE = FE \times \tan(51^\circ)
\]
\[
DE = 17 \times \tan(51^\circ)
\]
Using a calculator to find \( \tan(51^\circ) \approx 1.235 \):
\[
DE = 17 \times 1.235 \approx 21.0 \text{ meters}
\]
#### Multiple Choice Options:
- \( \) 10.7 m
- \( \) 13.2 m
- \( \) 13.8 m
- \( \) 21.0 m ✅
- \( \) 21.9 m
- \( \) 27.0 m
Thus, the length of \( DE \) is approximately **21.0 meters**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8bf67219-b61a-4356-beed-a1df9faed6c7%2Fb40b2c37-f7de-4d30-a7d1-a8e3adae8170%2F1cyyft_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Right Triangles: Finding the Length of an Unknown Side
In this exercise, we are given a right triangle \( \triangle FDE \) with a right angle at vertex \( E \), an angle of \( 51^\circ \) at vertex \( D \), and the length of \( FE = 17 \) meters.
We need to find the length of side \( DE \).
#### Triangle Details:
- **Angle \( \angle EFD \)** (adjacent to side \( DE \)): \( 51^\circ \)
- **Side \( FE \)** (adjacent to the \( 51^\circ \) angle): \( 17 \) meters
- **Right angle** at \( E \): \( \angle DEF = 90^\circ \)
#### Solution:
To find side \( DE \), we can use trigonometric ratios. Specifically, we use the tangent ratio in right triangles, which relates the opposite side to the adjacent side.
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Here:
- \( \theta = 51^\circ \)
- Adjacent side \( FE = 17 \) meters
- Opposite side \( DE \) (which we need to find)
\[
\tan(51^\circ) = \frac{DE}{FE}
\]
\[
DE = FE \times \tan(51^\circ)
\]
\[
DE = 17 \times \tan(51^\circ)
\]
Using a calculator to find \( \tan(51^\circ) \approx 1.235 \):
\[
DE = 17 \times 1.235 \approx 21.0 \text{ meters}
\]
#### Multiple Choice Options:
- \( \) 10.7 m
- \( \) 13.2 m
- \( \) 13.8 m
- \( \) 21.0 m ✅
- \( \) 21.9 m
- \( \) 27.0 m
Thus, the length of \( DE \) is approximately **21.0 meters**.
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