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
![**Title: Geometry Final**
**Question:**
The circle is centered on point \( P \). Points \( G, H, \) and \( I \) lie on the circumference. \(\angle GIH \) measures \( 21^\circ \). Find the measure of \(\angle GPH \).
**Diagram Description:**
- The diagram shows a circle centered on point \( P \).
- Points \( G, H, \) and \( I \) are positioned on the circumference of the circle.
- Line segments \( GP \), \( HP \), and \( IP \) are drawn from the points \( G, H, \) and \( I \) to the center point \( P \).
- Line segments \( GI \) and \( HI \) form the chord \( GIH \) on the circumference.
- The angle \(\angle GIH \) is marked at point \( I \) with a measurement of \( 21^\circ \).
**Explanation:**
In this problem, you need to determine the measure of \(\angle GPH\). This angle can be found using properties related to the circle and the given angle \(\angle GIH\).
**Steps to Solve:**
1. **Understand the Circle Property:**
The measure of the angle at the center (\(\angle GPH\)) of a circle is twice the measure of the angle on the circumference (\(\angle GIH\)) subtended by the same arc.
2. **Apply the Property:**
Given that \(\angle GIH = 21^\circ\), use the property to find:
\[
\angle GPH = 2 \times \angle GIH
\]
Substitute the given value:
\[
\angle GPH = 2 \times 21^\circ = 42^\circ
\]
**Answer:**
Thus, the measure of \(\angle GPH\) is \( 42^\circ \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F438dc635-e9b5-499b-83c3-a43302c8f333%2F66704d12-d7b4-4a9c-8e6c-81e3f9468f03%2F5w0609_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Geometry Final**
**Question:**
The circle is centered on point \( P \). Points \( G, H, \) and \( I \) lie on the circumference. \(\angle GIH \) measures \( 21^\circ \). Find the measure of \(\angle GPH \).
**Diagram Description:**
- The diagram shows a circle centered on point \( P \).
- Points \( G, H, \) and \( I \) are positioned on the circumference of the circle.
- Line segments \( GP \), \( HP \), and \( IP \) are drawn from the points \( G, H, \) and \( I \) to the center point \( P \).
- Line segments \( GI \) and \( HI \) form the chord \( GIH \) on the circumference.
- The angle \(\angle GIH \) is marked at point \( I \) with a measurement of \( 21^\circ \).
**Explanation:**
In this problem, you need to determine the measure of \(\angle GPH\). This angle can be found using properties related to the circle and the given angle \(\angle GIH\).
**Steps to Solve:**
1. **Understand the Circle Property:**
The measure of the angle at the center (\(\angle GPH\)) of a circle is twice the measure of the angle on the circumference (\(\angle GIH\)) subtended by the same arc.
2. **Apply the Property:**
Given that \(\angle GIH = 21^\circ\), use the property to find:
\[
\angle GPH = 2 \times \angle GIH
\]
Substitute the given value:
\[
\angle GPH = 2 \times 21^\circ = 42^\circ
\]
**Answer:**
Thus, the measure of \(\angle GPH\) is \( 42^\circ \).
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