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
Find the value of X and Y in the following right
![**Title: Solving Right Triangles**
**Introduction:**
In this lesson, we will explore the properties of a right-angled triangle and the methods to solve for unknown sides using trigonometric principles.
**Diagram Analysis:**
The diagram shows a right-angled triangle with the following properties:
- There is a right angle (\(90^\circ\)) at the vertex opposite the hypotenuse.
- One of the angles is \(30^\circ\).
- The hypotenuse has a length of \(6\sqrt{3}\).
- The sides opposite and adjacent to the \(30^\circ\) angle are labeled \(x\) and \(y\), respectively.
**Step-by-Step Solution:**
1. **Identify Known Values:**
- Hypotenuse (\(c\)): \(6\sqrt{3}\)
- Angle (\(\theta\)): \(30^\circ\)
2. **Use the Sine and Cosine Functions:**
The sine function for \(\theta = 30^\circ\):
\[
\sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{x}{6\sqrt{3}}
\]
We know that \(\sin(30^\circ) = \frac{1}{2}\), thus:
\[
\frac{1}{2} = \frac{x}{6\sqrt{3}}
\]
Solving for \(x\):
\[
x = \frac{1}{2} \times 6\sqrt{3} = 3\sqrt{3}
\]
The cosine function for \(\theta = 30^\circ\):
\[
\cos(30^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{y}{6\sqrt{3}}
\]
We know that \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\), thus:
\[
\frac{\sqrt{3}}{2} = \frac{y}{6\sqrt{3}}
\]
Solving for \(y\):
\[
y = \frac{\sqrt{3}}{2} \times 6\sqrt{3} = 3 \times 3 = 9](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4bb5a6dd-97cd-4d93-a8e0-dafecb13cdf7%2F6c8fe48f-a8fa-4fa2-9219-bb2cd42edaad%2Felw0g1d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Solving Right Triangles**
**Introduction:**
In this lesson, we will explore the properties of a right-angled triangle and the methods to solve for unknown sides using trigonometric principles.
**Diagram Analysis:**
The diagram shows a right-angled triangle with the following properties:
- There is a right angle (\(90^\circ\)) at the vertex opposite the hypotenuse.
- One of the angles is \(30^\circ\).
- The hypotenuse has a length of \(6\sqrt{3}\).
- The sides opposite and adjacent to the \(30^\circ\) angle are labeled \(x\) and \(y\), respectively.
**Step-by-Step Solution:**
1. **Identify Known Values:**
- Hypotenuse (\(c\)): \(6\sqrt{3}\)
- Angle (\(\theta\)): \(30^\circ\)
2. **Use the Sine and Cosine Functions:**
The sine function for \(\theta = 30^\circ\):
\[
\sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{x}{6\sqrt{3}}
\]
We know that \(\sin(30^\circ) = \frac{1}{2}\), thus:
\[
\frac{1}{2} = \frac{x}{6\sqrt{3}}
\]
Solving for \(x\):
\[
x = \frac{1}{2} \times 6\sqrt{3} = 3\sqrt{3}
\]
The cosine function for \(\theta = 30^\circ\):
\[
\cos(30^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{y}{6\sqrt{3}}
\]
We know that \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\), thus:
\[
\frac{\sqrt{3}}{2} = \frac{y}{6\sqrt{3}}
\]
Solving for \(y\):
\[
y = \frac{\sqrt{3}}{2} \times 6\sqrt{3} = 3 \times 3 = 9
Transcribed Image Text:### Right Triangle with a 30° Angle
The diagram shown is a right triangle with one of its angles measuring 30 degrees. The triangle includes the following components:
1. **Right Angle**: The triangle has one 90° angle, indicated by the small square on the left-hand side of the triangle.
2. **30° Angle**: One of the acute angles in the triangle measures 30°, as labeled on the diagram.
3. **Legs of the Triangle**:
- The leg adjacent to the 30° angle is labeled as **x**.
- The leg opposite the 30° angle is labeled as **y**.
4. **Hypotenuse**: The hypotenuse, which is the side opposite the right angle and the longest side of the triangle, is labeled as \(5\sqrt{3}\).
### Explanation of the Diagram
In trigonometry, a 30-60-90 triangle has a standardized ratio for the lengths of its sides. When the shortest side (adjacent to the 30° angle) is known, the lengths of the other sides can be determined using this ratio:
- Hypotenuse: \( 2 \times (\text{shortest side}) \)
- The side opposite the 30° angle is half of the hypotenuse.
- The side opposite the 60° angle (next to the 30° angle) is \( \sqrt{3} \times (\text{shortest side}) \).
Given that the hypotenuse (\(5\sqrt{3}\)) relates to this standard ratio, the triangle's measurements conform to these principles.
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