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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![### Solving a Right Triangle
In this educational example, we are given a right triangle with one angle measuring \(45^\circ\), a right angle, and one side length labeled as \(35\). We are asked to find the length of the side opposite the \(45^\circ\) angle, denoted as \(x\).
#### Diagram Explanation:
- **Triangle:** The diagram is a right triangle.
- **Angles:** The triangle includes a \(45^\circ\) angle and a right angle (marked with the small red square in the diagram).
- **Sides:**
- The side opposite the \(45^\circ\) angle is labeled \(x\).
- The hypotenuse (the side opposite the right angle) is specified as \(35\).
#### Steps to Solve for \(x\):
1. **Identify the Triangle Type:**
This triangle is a special type of right triangle known as an isosceles right triangle (45°-45°-90° triangle).
2. **Use the Properties of a 45°-45°-90° Triangle:**
In a 45°-45°-90° triangle, the sides have a specific ratio relative to the hypotenuse.
- The sides opposite the 45° angles are equal.
- Each side (leg) is \(\frac{\sqrt{2}}{2}\) times the hypotenuse.
3. **Calculate \(x\):**
Given the hypotenuse is 35 units,
\[
x = 35 \times \frac{\sqrt{2}}{2}
\]
Simplifying further,
\[
x = 35 \times \frac{\sqrt{2}}{2} = 35 \div \sqrt{2} \approx 24.75
\]
#### Solution:
- The length \(x\) is approximately \(24.75\).
#### Conclusion:
By recognizing the properties of a 45°-45°-90° triangle, you can quickly determine the lengths of the sides given one side. This knowledge is particularly useful when dealing with right triangles in various geometric and trigonometric problems.
**Box for Final Answer:**
\[
x = \boxed{24.75}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7f3d50e5-a330-4a3e-9ccb-00c03d665fb0%2Fd364cf6b-6841-486c-9e42-22d1dbee32a2%2Favu50ak_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving a Right Triangle
In this educational example, we are given a right triangle with one angle measuring \(45^\circ\), a right angle, and one side length labeled as \(35\). We are asked to find the length of the side opposite the \(45^\circ\) angle, denoted as \(x\).
#### Diagram Explanation:
- **Triangle:** The diagram is a right triangle.
- **Angles:** The triangle includes a \(45^\circ\) angle and a right angle (marked with the small red square in the diagram).
- **Sides:**
- The side opposite the \(45^\circ\) angle is labeled \(x\).
- The hypotenuse (the side opposite the right angle) is specified as \(35\).
#### Steps to Solve for \(x\):
1. **Identify the Triangle Type:**
This triangle is a special type of right triangle known as an isosceles right triangle (45°-45°-90° triangle).
2. **Use the Properties of a 45°-45°-90° Triangle:**
In a 45°-45°-90° triangle, the sides have a specific ratio relative to the hypotenuse.
- The sides opposite the 45° angles are equal.
- Each side (leg) is \(\frac{\sqrt{2}}{2}\) times the hypotenuse.
3. **Calculate \(x\):**
Given the hypotenuse is 35 units,
\[
x = 35 \times \frac{\sqrt{2}}{2}
\]
Simplifying further,
\[
x = 35 \times \frac{\sqrt{2}}{2} = 35 \div \sqrt{2} \approx 24.75
\]
#### Solution:
- The length \(x\) is approximately \(24.75\).
#### Conclusion:
By recognizing the properties of a 45°-45°-90° triangle, you can quickly determine the lengths of the sides given one side. This knowledge is particularly useful when dealing with right triangles in various geometric and trigonometric problems.
**Box for Final Answer:**
\[
x = \boxed{24.75}
\]
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