If the centroid of triangle ABC is G and BG = 9 cm, then what will be the length (in cm) of median BE?

Options

A. 12
B. 14
C. 15
D. 13.5

Answer

To find the length of the median BE in triangle ABC, given that the centroid G divides the median in the ratio of 2:1, we can follow these steps:

### Step 1: Understand the properties of the centroid
The centroid (G) of a triangle divides each median into two segments: the segment from the vertex to the centroid (in this case, BG) and the segment from the centroid to the midpoint of the opposite side (in this case, GE). The ratio of these segments is always 2:1, where the longer segment is from the vertex to the centroid.

### Step 2: Set up the relationship
Given that BG = 9 cm, we can denote:
- BG = 2x (the longer segment)
- GE = x (the shorter segment)

From the ratio of the segments, we have:
\[ BG : GE = 2 : 1 \]

### Step 3: Calculate the length of GE
Since BG = 2x and we know BG = 9 cm, we can set up the equation:
\[ 2x = 9 \]

To find x, we solve for x:
\[ x = \frac{9}{2} = 4.5 \text{ cm} \]

### Step 4: Find the length of the median BE
The total length of the median BE is the sum of BG and GE:
\[ BE = BG + GE = 9 + 4.5 = 13.5 \text{ cm} \]

### Final Answer
Thus, the length of median BE is **13.5 cm**.

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