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
![The image contains a hand-drawn right triangle, labeled with an interior 60° angle, and some side lengths. Below is a detailed description suitable for an educational website.
---
### Understanding Right Triangles with 60° Angles
In this example, we have a right triangle with one angle measuring 60°. The diagram is annotated with the length of the hypotenuse and the variables representing the other sides.
#### Diagram Description:
1. **Hypotenuse:** The side opposite the right-angle, labeled as \( 5\sqrt{3} \).
2. **Angle:** One of the non-right angles is marked as 60°.
3. **Sides:**
- The side opposite the 60° angle is labeled \( y \).
- The side adjacent to the 60° angle is labeled \( x \).
According to the properties of a 30°-60°-90° triangle:
- The hypotenuse is always twice the shorter leg.
- The longer leg opposite the 60° angle is \( \sqrt{3} \) times the shorter leg.
#### Solving for \( x \) and \( y \):
1. Given the hypotenuse \( 5\sqrt{3} \), we can find \( x \) and \( y \):
- **Shorter leg (x):** Since in a 30°-60°-90° triangle, the hypotenuse is twice the shorter leg,
\[
x = \frac{hypotenuse}{2} = \frac{5\sqrt{3}}{2} = \frac{5\sqrt{3}}{2}
\]
- **Longer leg (y):** This is \( \sqrt{3} \) times the shorter leg,
\[
y = \sqrt{3} \times x = \sqrt{3} \times \frac{5\sqrt{3}}{2} = \frac{5 \times 3}{2} = \frac{15}{2} = 7.5
\]
#### Conclusion:
In this triangle:
- \( x = \frac{5\sqrt{3}}{2} \)
- \( y = 7.5 \)
These calculations help in understanding the properties of 30°-60°-90° triangles and solving for missing side lengths when given one side.
---
This explanation provides](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0f6f557d-9e1e-46fd-a63e-879d74571691%2F9c4dcc97-80c5-4c84-a822-f8fced69a0c5%2Fvpos9d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains a hand-drawn right triangle, labeled with an interior 60° angle, and some side lengths. Below is a detailed description suitable for an educational website.
---
### Understanding Right Triangles with 60° Angles
In this example, we have a right triangle with one angle measuring 60°. The diagram is annotated with the length of the hypotenuse and the variables representing the other sides.
#### Diagram Description:
1. **Hypotenuse:** The side opposite the right-angle, labeled as \( 5\sqrt{3} \).
2. **Angle:** One of the non-right angles is marked as 60°.
3. **Sides:**
- The side opposite the 60° angle is labeled \( y \).
- The side adjacent to the 60° angle is labeled \( x \).
According to the properties of a 30°-60°-90° triangle:
- The hypotenuse is always twice the shorter leg.
- The longer leg opposite the 60° angle is \( \sqrt{3} \) times the shorter leg.
#### Solving for \( x \) and \( y \):
1. Given the hypotenuse \( 5\sqrt{3} \), we can find \( x \) and \( y \):
- **Shorter leg (x):** Since in a 30°-60°-90° triangle, the hypotenuse is twice the shorter leg,
\[
x = \frac{hypotenuse}{2} = \frac{5\sqrt{3}}{2} = \frac{5\sqrt{3}}{2}
\]
- **Longer leg (y):** This is \( \sqrt{3} \) times the shorter leg,
\[
y = \sqrt{3} \times x = \sqrt{3} \times \frac{5\sqrt{3}}{2} = \frac{5 \times 3}{2} = \frac{15}{2} = 7.5
\]
#### Conclusion:
In this triangle:
- \( x = \frac{5\sqrt{3}}{2} \)
- \( y = 7.5 \)
These calculations help in understanding the properties of 30°-60°-90° triangles and solving for missing side lengths when given one side.
---
This explanation provides
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