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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Ratios
A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.
Trigonometric Ratios
Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.
Question
![The image presents a geometrical problem involving a circle with tangents and an external angle. Here is a detailed description of the elements in the image:
1. **Geometry of the Diagram**:
- A circle is depicted with two tangents drawn from an external point, forming an angle outside the circle.
- The angle formed outside the circle by the tangents is labeled as \(118^\circ\).
- The angle between one of the tangents and the segment connecting the point of tangency with the center of the circle is labeled \(24^\circ\).
- The angle \(x^\circ\) is the central angle of the arc that the external tangents subtend on the circle.
2. **Problem Statement**:
- The task is to find the value of the angle \(x\).
3. **Formula and Calculation**:
- The exterior angle (formed by the tangents and the circle) is supplementary to the sum of the interior opposite angles (relevant here, the central angle \(x^\circ\)).
- An important property of a circle is that the sum of the opposite angles of a cyclic quadrilateral is \(180^\circ\). Since the tangents create a triangle with the center of the circle, we focus on this relation.
- The external angle outside the circle is equal to half the difference of the intercepted arcs or the relevant angles in the circle.
Given these points:
\[ \text{External Angle} = \frac{1}{2} \left| \text{Difference of intercepted arcs} \right| \]
Here, both intercepted arcs refer to \( x \).
4. **Finding the Value of \( x \)**:
- Using the angle formula: Exterior angle \( 118^\circ = \frac{1}{2} \times (\text{Angle facing intercepted arcs}) \)
- Therefore, \(118^\circ = \frac{1}{2} \times x^\circ\)
Solving for \( x \):
\[ 118^\circ \times 2 = x^\circ \]
\[ x = 236^\circ \]
Thus, the value of \( x \) is \( 236^\circ \).
This is now presented in an interactive format where students can input their calculated value for \( x \).
**Interactive Component**:
```
Find the value of \( x \).
\[ x = \boxed{} \]
```
By solving, the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F251d803b-062a-485c-ac85-18fab7c31e4e%2Fa73513be-6538-4c9c-b965-96765b8a8b65%2Fh3ivu3r_processed.png&w=3840&q=75)
Transcribed Image Text:The image presents a geometrical problem involving a circle with tangents and an external angle. Here is a detailed description of the elements in the image:
1. **Geometry of the Diagram**:
- A circle is depicted with two tangents drawn from an external point, forming an angle outside the circle.
- The angle formed outside the circle by the tangents is labeled as \(118^\circ\).
- The angle between one of the tangents and the segment connecting the point of tangency with the center of the circle is labeled \(24^\circ\).
- The angle \(x^\circ\) is the central angle of the arc that the external tangents subtend on the circle.
2. **Problem Statement**:
- The task is to find the value of the angle \(x\).
3. **Formula and Calculation**:
- The exterior angle (formed by the tangents and the circle) is supplementary to the sum of the interior opposite angles (relevant here, the central angle \(x^\circ\)).
- An important property of a circle is that the sum of the opposite angles of a cyclic quadrilateral is \(180^\circ\). Since the tangents create a triangle with the center of the circle, we focus on this relation.
- The external angle outside the circle is equal to half the difference of the intercepted arcs or the relevant angles in the circle.
Given these points:
\[ \text{External Angle} = \frac{1}{2} \left| \text{Difference of intercepted arcs} \right| \]
Here, both intercepted arcs refer to \( x \).
4. **Finding the Value of \( x \)**:
- Using the angle formula: Exterior angle \( 118^\circ = \frac{1}{2} \times (\text{Angle facing intercepted arcs}) \)
- Therefore, \(118^\circ = \frac{1}{2} \times x^\circ\)
Solving for \( x \):
\[ 118^\circ \times 2 = x^\circ \]
\[ x = 236^\circ \]
Thus, the value of \( x \) is \( 236^\circ \).
This is now presented in an interactive format where students can input their calculated value for \( x \).
**Interactive Component**:
```
Find the value of \( x \).
\[ x = \boxed{} \]
```
By solving, the
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