M is the centroid of ADEF. D DM = 8 MJ = 2y L E EM = 6 K FM = 2x F What is EL? O A. 9 о в 4 O C. 3 о D. 12

Elementary Geometry For College Students, 7e
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ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
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Question 4

### Centroid of a Triangle

In the diagram, point \( M \) is the centroid of the triangle \( \Delta DEF \).

#### Details of the Diagram
- The triangle \( DEF \) has vertices \( D \), \( E \), and \( F \).
- \( M \) is the centroid, which means it is the point where the three medians of the triangle intersect.
- A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.

#### Explanation of the Medians
- \( DM \) is a median, and \( DM = 8 \).
- \( MJ \) is a segment of the median \( DM \), and \( MJ = 2y \).

- \( EM \) is another median, and \( EM = 6 \).

- \( FM \) is the third median, and \( FM = 2x \).

#### Finding EL
We are tasked with finding the length of \( EL \).

**Given Options:**
- A. 9
- B. 4
- C. 3
- D. 12

Given that \( M \) is the centroid, it divides each median into two segments, with the longer segment being twice the length of the shorter segment. Therefore:
1. \( DM = DJ + JM \)
   - Since \( M \) is the centroid and divides \( DM \) in a 2:1 ratio:
   - \( DJ = \frac{2}{3} \times DM = \frac{2}{3} \times 8 = 16/3 \)
   - \( JM = \frac{1}{3} \times DM = \frac{1}{3} \times 8 = 8/3 \)
2. \( EM = 6 \)
3. \( FM = 2x \)

To find \( EL \):
Since \( M \) is the centroid, \( EL = 3 \)

Correct answer: **C. 3**
Transcribed Image Text:### Centroid of a Triangle In the diagram, point \( M \) is the centroid of the triangle \( \Delta DEF \). #### Details of the Diagram - The triangle \( DEF \) has vertices \( D \), \( E \), and \( F \). - \( M \) is the centroid, which means it is the point where the three medians of the triangle intersect. - A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. #### Explanation of the Medians - \( DM \) is a median, and \( DM = 8 \). - \( MJ \) is a segment of the median \( DM \), and \( MJ = 2y \). - \( EM \) is another median, and \( EM = 6 \). - \( FM \) is the third median, and \( FM = 2x \). #### Finding EL We are tasked with finding the length of \( EL \). **Given Options:** - A. 9 - B. 4 - C. 3 - D. 12 Given that \( M \) is the centroid, it divides each median into two segments, with the longer segment being twice the length of the shorter segment. Therefore: 1. \( DM = DJ + JM \) - Since \( M \) is the centroid and divides \( DM \) in a 2:1 ratio: - \( DJ = \frac{2}{3} \times DM = \frac{2}{3} \times 8 = 16/3 \) - \( JM = \frac{1}{3} \times DM = \frac{1}{3} \times 8 = 8/3 \) 2. \( EM = 6 \) 3. \( FM = 2x \) To find \( EL \): Since \( M \) is the centroid, \( EL = 3 \) Correct answer: **C. 3**
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