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
Concept explainers
Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
Topic Video
Question
Find the value of X
Transcribed Image Text:This image is a diagram of a circle containing an inscribed triangle. The circle has an angle labeled 170 degrees, formed by two radii and a chord of the circle. Inside the triangle, two sides are labeled as 6 (units).
Key elements of the diagram include:
- A circle with an inscribed triangle and a chord.
- The vertex angle outside the circle is given as 170 degrees.
- Two of the triangle's sides within the circle are labeled with a length of 6 units each.
- There is an unknown angle, labeled as \( x \) degrees, opposite the chord of the circle.
To solve for the unknown \( x \) degrees, one can use the properties of circles, triangles, and inscribed angles. In particular, knowing that the sum of angles in any triangle is 180 degrees and applying properties of the specific types of triangles formed with circles (like isosceles triangles with equal radii) can help derive \( x \).
![### Understanding Circle Segment Geometry
In the image shown, we have a circle with some key geometric features, which we need to analyze to solve for the unknown angle \( x \).
#### Description of the Diagram:
1. **Circle:** The basic shape in the diagram is a circle.
2. **Angles:** Inside the circle, there are two given angles:
- An angle of **70°** at one segment of the circle.
- An angle of **220°** on the opposite segment of the circle.
3. **Chord:** The circle contains a chord labeled with the length **8**. Below this chord is another angle labeled as \( x \).
#### Explanation and Solution:
To understand and solve for \( x \), we will use the following key concepts in circle geometry:
1. **Sum of Angles in a Circle:**
- A full circle makes up **360°**.
- The sum of all angles around a point inside a circle also adds up to **360°**.
Given that one segment of the circle has an angle of 70° and the remaining segment must sum up to 360° when combined with 70°, the remaining segment angle is:
\[ 360° - 220° = 140° \]
This leaves the angle opposite to \( x \) as 140° since the sum of angles at a point (360°) deducts the known 220° section.
2. **Using the Alternate Segment Theorem:**
- According to this theorem, the angle between the tangent and the chord at the point of contact is equal to the angle in the alternate segment.
However, since we don't have a tangent here, we will rely on segment geometry principles and the fact that angle measures in respective segments must add up to complete the circumference angle.
Hence, the angle \( x \) (vertical opposite) should be derived from subtraction:
\[ x = 360° - (70° + 220°) = 70° \]
Thus, the unknown angle \( x \) is **70°**.
In summary, recognizing the complementary properties of angles in a circle can help solve for unknown angles effectively. Further grasp of geometric theorems can provide deeper insights and ease in solving complex problems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4bb5a6dd-97cd-4d93-a8e0-dafecb13cdf7%2Fb8fe0307-3cc0-441e-a947-0e99663a9961%2Fmszk4cc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Circle Segment Geometry
In the image shown, we have a circle with some key geometric features, which we need to analyze to solve for the unknown angle \( x \).
#### Description of the Diagram:
1. **Circle:** The basic shape in the diagram is a circle.
2. **Angles:** Inside the circle, there are two given angles:
- An angle of **70°** at one segment of the circle.
- An angle of **220°** on the opposite segment of the circle.
3. **Chord:** The circle contains a chord labeled with the length **8**. Below this chord is another angle labeled as \( x \).
#### Explanation and Solution:
To understand and solve for \( x \), we will use the following key concepts in circle geometry:
1. **Sum of Angles in a Circle:**
- A full circle makes up **360°**.
- The sum of all angles around a point inside a circle also adds up to **360°**.
Given that one segment of the circle has an angle of 70° and the remaining segment must sum up to 360° when combined with 70°, the remaining segment angle is:
\[ 360° - 220° = 140° \]
This leaves the angle opposite to \( x \) as 140° since the sum of angles at a point (360°) deducts the known 220° section.
2. **Using the Alternate Segment Theorem:**
- According to this theorem, the angle between the tangent and the chord at the point of contact is equal to the angle in the alternate segment.
However, since we don't have a tangent here, we will rely on segment geometry principles and the fact that angle measures in respective segments must add up to complete the circumference angle.
Hence, the angle \( x \) (vertical opposite) should be derived from subtraction:
\[ x = 360° - (70° + 220°) = 70° \]
Thus, the unknown angle \( x \) is **70°**.
In summary, recognizing the complementary properties of angles in a circle can help solve for unknown angles effectively. Further grasp of geometric theorems can provide deeper insights and ease in solving complex problems.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, geometry and related others by exploring similar questions and additional content below.Recommended textbooks for you
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning