24. Round to the nearest tenth if necessary. 1 13. Point P is the centroid for AMNO. Calculate RP if RN M R S RP = 16 RP = 9 RP = 12 ce ing C P N
24. Round to the nearest tenth if necessary. 1 13. Point P is the centroid for AMNO. Calculate RP if RN M R S RP = 16 RP = 9 RP = 12 ce ing C P N
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
![### Geometry Problem: Finding Segment Lengths in a Triangle with a Centroid
#### Problem Statement:
Point \( P \) is the centroid for \( \Delta MNO \). Calculate \( RP \) if \( RN = 24 \). Round to the nearest tenth if necessary.
#### Given:
- \( \Delta MNO \) with centroid \( P \)
- \( RN = 24 \)
#### Diagram Explanation:
The diagram represents a triangle \( MNO \) with the centroid \( P \). Lines from the vertices \( M \), \( N \), and \( O \) intersect at the centroid \( P \). Points \( R \), \( S \), and \( Q \) are midpoints of sides \( MO \), \( MN \), and \( ON \) respectively.
The centroid \( P \) divides each median into a ratio of 2:1, where the centroid is twice as close to each midpoint as it is to each corresponding vertex.
#### Calculation:
Given \( RN = 24 \):
- Since \( P \) is the centroid, \( P \) divides the median \( RN \) in the ratio 2:1.
- If \( RN = 24 \), then \( RP \) is one-third of this distance (since \( P \) divides \( RN \) into \( RP \) and \( PN \) in the ratio 2:1).
**Mathematical Calculation:**
\[ RP = \frac{1}{3} \times 24 = 8 \]
#### Options:
- \( RP = 16 \) (Incorrect)
- \( RP = 9 \) (Incorrect)
- \( RP = 12 \) (Incorrect)
- \( RP = 8 \) (Correct)
Thus, the correct answer is:
\[ RP = 8 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c2f64fc-e07a-43b3-a4b0-6f03d401f48c%2F7851556d-c20b-4abe-92e2-ad1349cb8bb0%2Fvwi2ox_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Geometry Problem: Finding Segment Lengths in a Triangle with a Centroid
#### Problem Statement:
Point \( P \) is the centroid for \( \Delta MNO \). Calculate \( RP \) if \( RN = 24 \). Round to the nearest tenth if necessary.
#### Given:
- \( \Delta MNO \) with centroid \( P \)
- \( RN = 24 \)
#### Diagram Explanation:
The diagram represents a triangle \( MNO \) with the centroid \( P \). Lines from the vertices \( M \), \( N \), and \( O \) intersect at the centroid \( P \). Points \( R \), \( S \), and \( Q \) are midpoints of sides \( MO \), \( MN \), and \( ON \) respectively.
The centroid \( P \) divides each median into a ratio of 2:1, where the centroid is twice as close to each midpoint as it is to each corresponding vertex.
#### Calculation:
Given \( RN = 24 \):
- Since \( P \) is the centroid, \( P \) divides the median \( RN \) in the ratio 2:1.
- If \( RN = 24 \), then \( RP \) is one-third of this distance (since \( P \) divides \( RN \) into \( RP \) and \( PN \) in the ratio 2:1).
**Mathematical Calculation:**
\[ RP = \frac{1}{3} \times 24 = 8 \]
#### Options:
- \( RP = 16 \) (Incorrect)
- \( RP = 9 \) (Incorrect)
- \( RP = 12 \) (Incorrect)
- \( RP = 8 \) (Correct)
Thus, the correct answer is:
\[ RP = 8 \]
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