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: An Example
#### Diagram Explanation:
The diagram presented is of a right-angled triangle, which includes the following elements:
1. **Right Angle**: The diagram has a right angle represented by a small square in one of its corners.
2. **Sides**:
- One side (`base`) of the triangle is labeled with a length of `4`.
- The hypotenuse (the side opposite the right angle) is labeled with a length of `5`.
3. **Unknown Angle**:
- There is an angle marked with a `?` symbol, indicating that its measure needs to be determined.
This right triangle provides a realistic example for understanding fundamental geometry principles such as the Pythagorean theorem and trigonometric ratios in right triangles.
#### Key Concepts:
1. **Pythagorean Theorem**:
- The relationship between the lengths of the sides of a right-angled triangle is given by:
\[ a^2 + b^2 = c^2 \]
- In this diagram:
- \( a = 4 \)
- \( c = 5 \)
Using these values, you can check that the other perpendicular side, \( b \), remains consistent with the theorem.
2. **Trigonometric Ratios**:
- To find the unknown angle (`?`), trigonometric ratios such as sine, cosine, and tangent can be applied:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
- For the given values, to find the angle `?`:
\[ \tan(\theta) = \frac{4}{b} = \frac{4}{\sqrt{5^2 - 4^2}} = \frac{4}{3} \]
\[ \theta = \tan^{-1} (\frac{4}{3}) \]
Thus, using the given right triangle's sides with the Pythagorean theorem and trigonometric ratios, one can determine the measure of the unknown angle `?`.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2f3e5dad-3eef-4af4-8432-92621aeb9e2a%2Feb426641-9e93-4fd0-847d-1cd4323f92a0%2F12nzoxg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Right Triangles: An Example
#### Diagram Explanation:
The diagram presented is of a right-angled triangle, which includes the following elements:
1. **Right Angle**: The diagram has a right angle represented by a small square in one of its corners.
2. **Sides**:
- One side (`base`) of the triangle is labeled with a length of `4`.
- The hypotenuse (the side opposite the right angle) is labeled with a length of `5`.
3. **Unknown Angle**:
- There is an angle marked with a `?` symbol, indicating that its measure needs to be determined.
This right triangle provides a realistic example for understanding fundamental geometry principles such as the Pythagorean theorem and trigonometric ratios in right triangles.
#### Key Concepts:
1. **Pythagorean Theorem**:
- The relationship between the lengths of the sides of a right-angled triangle is given by:
\[ a^2 + b^2 = c^2 \]
- In this diagram:
- \( a = 4 \)
- \( c = 5 \)
Using these values, you can check that the other perpendicular side, \( b \), remains consistent with the theorem.
2. **Trigonometric Ratios**:
- To find the unknown angle (`?`), trigonometric ratios such as sine, cosine, and tangent can be applied:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
- For the given values, to find the angle `?`:
\[ \tan(\theta) = \frac{4}{b} = \frac{4}{\sqrt{5^2 - 4^2}} = \frac{4}{3} \]
\[ \theta = \tan^{-1} (\frac{4}{3}) \]
Thus, using the given right triangle's sides with the Pythagorean theorem and trigonometric ratios, one can determine the measure of the unknown angle `?`.
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