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
![### Right Triangle and Trigonometry
In the provided image, we have a diagram of a right triangle with one of its non-right angles labeled as 60°. The sides of the triangle are marked with variables and lengths, and there's a brief handwritten note.
#### Details of the Triangle:
1. **Hypotenuse:** Labeled as \( 5\sqrt{3} \)
2. **Opposite Side to 60° (y):** This is labeled as 'y'.
3. **Adjacent Side to 60° (x):** This is labeled as 'x'.
4. **Angle:** One of the angles is 60°.
5. **Right Angle:** As denoted by the right angle box at the intersection of sides 'x' and 'y'.
#### Equations and Calculations:
- There's a note with the equation: \( \text{hypotenuse} = \text{leg} \cdot \sqrt{3} \).
This suggests the use of properties of 30-60-90 triangles where:
- The length of the hypotenuse is \( 2 \times \text{short side} \).
- The length of the longer leg (adjacent side to the 60° angle) is \( \sqrt{3} \times \text{short side} \).
Given:
- \( \text{Hypotenuse} = 5\sqrt{3} \)
In a 30-60-90 triangle:
- \( y \) (short leg, opposite to 30°) would be \( 5 \)
- \( x \) (longer leg, adjacent to 60°) would be \( 5\sqrt{3} \)
However, we need to work through the calculations using trigonometric identities or properties for precise confirmation.
**Visualizing the Diagram:**
- There’s a right triangle.
- Hypotenuse is the side opposite the right angle.
- A 60° angle is opposite side ‘y’.
- The right angle is between sides 'x' and 'y'.
By using the trigonometric functions and ratios for a 30-60-90 triangle:
\[
\text{Hypotenuse} = 2 \times \text{short leg}
\]
\[
\text{Longer leg} = \sqrt{3} \times \text{short leg}
\]
Thus, we can conclude:
- The short leg](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0f6f557d-9e1e-46fd-a63e-879d74571691%2F05688b24-1de3-419e-a568-3dc25edb441b%2Fi2miyo9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Right Triangle and Trigonometry
In the provided image, we have a diagram of a right triangle with one of its non-right angles labeled as 60°. The sides of the triangle are marked with variables and lengths, and there's a brief handwritten note.
#### Details of the Triangle:
1. **Hypotenuse:** Labeled as \( 5\sqrt{3} \)
2. **Opposite Side to 60° (y):** This is labeled as 'y'.
3. **Adjacent Side to 60° (x):** This is labeled as 'x'.
4. **Angle:** One of the angles is 60°.
5. **Right Angle:** As denoted by the right angle box at the intersection of sides 'x' and 'y'.
#### Equations and Calculations:
- There's a note with the equation: \( \text{hypotenuse} = \text{leg} \cdot \sqrt{3} \).
This suggests the use of properties of 30-60-90 triangles where:
- The length of the hypotenuse is \( 2 \times \text{short side} \).
- The length of the longer leg (adjacent side to the 60° angle) is \( \sqrt{3} \times \text{short side} \).
Given:
- \( \text{Hypotenuse} = 5\sqrt{3} \)
In a 30-60-90 triangle:
- \( y \) (short leg, opposite to 30°) would be \( 5 \)
- \( x \) (longer leg, adjacent to 60°) would be \( 5\sqrt{3} \)
However, we need to work through the calculations using trigonometric identities or properties for precise confirmation.
**Visualizing the Diagram:**
- There’s a right triangle.
- Hypotenuse is the side opposite the right angle.
- A 60° angle is opposite side ‘y’.
- The right angle is between sides 'x' and 'y'.
By using the trigonometric functions and ratios for a 30-60-90 triangle:
\[
\text{Hypotenuse} = 2 \times \text{short leg}
\]
\[
\text{Longer leg} = \sqrt{3} \times \text{short leg}
\]
Thus, we can conclude:
- The short leg
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