If the centroid of `DeltaMNP` is G and MG = 5cm, then the median `bar(MD)` in cm is

Options

A. 7.5
B. 7
C. 6.5
D. 6

Answer

To find the length of the median \( \bar{MD} \) in triangle \( \Delta MNP \) given that the centroid \( G \) divides the median \( \bar{MD} \) in the ratio \( 2:1 \) and \( MG = 5 \) cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the properties of the centroid**: The centroid \( G \) of a triangle divides each median into two segments, where the segment connecting the vertex to the centroid is twice as long as the segment connecting the centroid to the midpoint of the opposite side. This means that if \( MG \) is the segment from vertex \( M \) to centroid \( G \) and \( GD \) is the segment from centroid \( G \) to midpoint \( D \) of side \( NP \), then:
\[ \frac{MG}{GD} = \frac{2}{1} \]

2. **Set up the relationship**: Let \( MG = 5 \) cm. According to the ratio \( 2:1 \):
\[ MG = 2 \times GD \]
Therefore, we can express \( GD \) in terms of \( MG \):
\[ GD = \frac{MG}{2} = \frac{5}{2} \text{ cm} = 2.5 \text{ cm} \]

3. **Calculate the total length of the median \( \bar{MD} \)**: The total length of the median \( \bar{MD} \) is the sum of \( MG \) and \( GD \):
\[ MD = MG + GD = 5 \text{ cm} + 2.5 \text{ cm} = 7.5 \text{ cm} \]

4. **Conclusion**: Thus, the length of the median \( \bar{MD} \) is:
\[ \bar{MD} = 7.5 \text{ cm} \]

### Final Answer:
The median \( \bar{MD} \) is \( 7.5 \) cm.

---