5. In circle M, the central angle LMN measures (14x-48)° and inscribed angle LPN measures (5x+12)°. What is the measure of LN in degrees? 19v-48 5X+2
5. In circle M, the central angle LMN measures (14x-48)° and inscribed angle LPN measures (5x+12)°. What is the measure of LN in degrees? 19v-48 5X+2
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
100%
Answer question 5
![### Geometry Problem: Central and Inscribed Angles
**Problem Statement:**
5. In circle \(M\), the central angle \(\angle LMN\) measures \((14x - 48)^\circ\) and the inscribed angle \(\angle LPN\) measures \((5x + 12)^\circ\). What is the measure of arc \(\overset{\frown}{LN}\) in degrees?
**Illustration Description:**
The image depicts a diagram of a circle with center \(M\). Points \(L\), \(N\), and \(P\) are located on the circumference of the circle, creating two angles:
- \(\angle LMN\) is the central angle, indicated as \((14x - 48)^\circ\).
- \(\angle LPN\) is the inscribed angle, noted as \((5x + 12)^\circ\).
The objective is to determine the measure of arc \(\overset{\frown}{LN}\).
**Solution Steps:**
1. **Identify Relationship Between Central and Inscribed Angles:**
In a circle, the measure of an inscribed angle is half the measure of the intercepted arc or its corresponding central angle. Therefore, the relationship between the angles is given by:
\[
\angle LPN = \frac{1}{2} \angle LMN
\]
2. **Substitute Given Angle Measures:**
\[
5x + 12 = \frac{1}{2}(14x - 48)
\]
3. **Solve the Equation:**
Multiply both sides of the equation by 2 to eliminate the fraction:
\[
2(5x + 12) = 14x - 48
\]
Expand and simplify:
\[
10x + 24 = 14x - 48
\]
Rearrange the equation to isolate \(x\):
\[
24 + 48 = 14x - 10x
\]
\[
72 = 4x
\]
\[
x = 18
\]
4. **Determine the Measure of the Central Angle:**
Substitute \(x = 18\) back into the expression for \(\angle LMN\):
\[
\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F48b0c420-ba29-475d-827a-2f33ce024949%2F0094b978-3bbd-4c88-b572-00e09ff4382c%2F5xaoehi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Geometry Problem: Central and Inscribed Angles
**Problem Statement:**
5. In circle \(M\), the central angle \(\angle LMN\) measures \((14x - 48)^\circ\) and the inscribed angle \(\angle LPN\) measures \((5x + 12)^\circ\). What is the measure of arc \(\overset{\frown}{LN}\) in degrees?
**Illustration Description:**
The image depicts a diagram of a circle with center \(M\). Points \(L\), \(N\), and \(P\) are located on the circumference of the circle, creating two angles:
- \(\angle LMN\) is the central angle, indicated as \((14x - 48)^\circ\).
- \(\angle LPN\) is the inscribed angle, noted as \((5x + 12)^\circ\).
The objective is to determine the measure of arc \(\overset{\frown}{LN}\).
**Solution Steps:**
1. **Identify Relationship Between Central and Inscribed Angles:**
In a circle, the measure of an inscribed angle is half the measure of the intercepted arc or its corresponding central angle. Therefore, the relationship between the angles is given by:
\[
\angle LPN = \frac{1}{2} \angle LMN
\]
2. **Substitute Given Angle Measures:**
\[
5x + 12 = \frac{1}{2}(14x - 48)
\]
3. **Solve the Equation:**
Multiply both sides of the equation by 2 to eliminate the fraction:
\[
2(5x + 12) = 14x - 48
\]
Expand and simplify:
\[
10x + 24 = 14x - 48
\]
Rearrange the equation to isolate \(x\):
\[
24 + 48 = 14x - 10x
\]
\[
72 = 4x
\]
\[
x = 18
\]
4. **Determine the Measure of the Central Angle:**
Substitute \(x = 18\) back into the expression for \(\angle LMN\):
\[
\
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
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