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
![### Geometry Problem: Finding the Length of a Side
The image features a right triangle \( \triangle STU \). Below are the details and the diagram of the triangle:
**Diagram Description:**
- The triangle \( \triangle STU \) is depicted with:
- Angle \( \angle STU = 21^\circ \)
- \( ST = x \)
- \( TU = 5.5 \)
- \( \angle T = 90^\circ \)
**Purpose:**
To solve problems involving the sides and angles of right triangles, specifically focusing on trigonometric relationships.
**Explanation:**
Given:
1. A right triangle \( \triangle STU \)
2. \( \angle STU = 21^\circ \)
3. \( \text{Length of } TU = 5.5 \)
4. \( \text{Length of } ST = x \)
5. \( \angle T = 90^\circ \)
### Trigonometric Relationship:
To find the unknown length \( x \), we can use the trigonometric ratios, specifically the tangent (tan) function since we know an angle and the length of the side adjacent to it.
\[ \tan(21^\circ) = \frac{\text{opposite}}{\text{adjacent}} \]
\[ \tan(21^\circ) = \frac{ST}{TU} \]
\[ \tan(21^\circ) = \frac{x}{5.5} \]
From the equation, solve for \( x \):
\[ x = 5.5 \times \tan(21^\circ) \]
Using a calculator, find \( \tan(21^\circ) \) and then multiply by 5.5 to find the length of \( ST \).
This process demonstrates how to utilize the tangent function to determine unknown side lengths in right triangles using known angles and side lengths.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0f6f557d-9e1e-46fd-a63e-879d74571691%2Fa419d321-bb5c-4816-98a0-7b18968acb2a%2Fh9cq6at_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Geometry Problem: Finding the Length of a Side
The image features a right triangle \( \triangle STU \). Below are the details and the diagram of the triangle:
**Diagram Description:**
- The triangle \( \triangle STU \) is depicted with:
- Angle \( \angle STU = 21^\circ \)
- \( ST = x \)
- \( TU = 5.5 \)
- \( \angle T = 90^\circ \)
**Purpose:**
To solve problems involving the sides and angles of right triangles, specifically focusing on trigonometric relationships.
**Explanation:**
Given:
1. A right triangle \( \triangle STU \)
2. \( \angle STU = 21^\circ \)
3. \( \text{Length of } TU = 5.5 \)
4. \( \text{Length of } ST = x \)
5. \( \angle T = 90^\circ \)
### Trigonometric Relationship:
To find the unknown length \( x \), we can use the trigonometric ratios, specifically the tangent (tan) function since we know an angle and the length of the side adjacent to it.
\[ \tan(21^\circ) = \frac{\text{opposite}}{\text{adjacent}} \]
\[ \tan(21^\circ) = \frac{ST}{TU} \]
\[ \tan(21^\circ) = \frac{x}{5.5} \]
From the equation, solve for \( x \):
\[ x = 5.5 \times \tan(21^\circ) \]
Using a calculator, find \( \tan(21^\circ) \) and then multiply by 5.5 to find the length of \( ST \).
This process demonstrates how to utilize the tangent function to determine unknown side lengths in right triangles using known angles and side lengths.
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