If distance of centroid of triangle ABC from vertex A is 6 cm them find the length of median through point, A.

Options

A. (a) 7cm
B. (b) 9 cm
C. (c) 10cm
D. (d) 12cm

Answer

To solve the problem, we need to find the length of the median from vertex A to the centroid of triangle ABC, given that the distance from vertex A to the centroid is 6 cm.

### Step-by-Step Solution:

1. **Understanding the Centroid**:
The centroid (G) of a triangle is the point where the three medians intersect. It divides each median into two segments, with the segment connecting the vertex to the centroid being twice as long as the segment connecting the centroid to the midpoint of the opposite side. Therefore, the centroid divides each median in the ratio 2:1.

2. **Given Information**:
We are given that the distance from vertex A to the centroid G is 6 cm. This means AG = 6 cm.

3. **Using the Ratio of the Median**:
Since the centroid divides the median in the ratio 2:1, we can denote the length of the median from vertex A to the midpoint of side BC as AM. Here, AG (the segment from A to G) is 2 parts of the total median AM, and GM (the segment from G to M) is 1 part.

Therefore, we can express the lengths as:
- AG = 2x (where x is the length of GM)
- GM = x

Since AG = 6 cm, we can set up the equation:
\[ 2x = 6 \text{ cm} \]

4. **Solving for x**:
To find x, we divide both sides of the equation by 2:
\[ x = \frac{6}{2} = 3 \text{ cm} \]

5. **Finding the Total Length of the Median**:
The total length of the median AM is the sum of AG and GM:
\[ AM = AG + GM = 6 \text{ cm} + 3 \text{ cm} = 9 \text{ cm} \]

### Final Answer:
The length of the median through point A is **9 cm**.