In a `DeltaABC`, the medians AD, BE and CF passes through G. If FG = 3.5 cm. find GC.

Options

A. 6 cm
B. 7 cm
C. 8 cm
D. 9 cm

Answer

To solve the problem, we need to find the length of segment GC in triangle ABC, where G is the centroid and FG is given as 3.5 cm.

### Step-by-Step Solution:

1. **Understanding the Centroid**:
The centroid (G) of a triangle divides each median in the ratio of 2:1. This means that the segment from the vertex to the centroid (e.g., AG) is twice the length of the segment from the centroid to the midpoint of the opposite side (e.g., GC).

2. **Identifying the Segments**:
In this case, FG is given as 3.5 cm. Since FG is the segment from the centroid G to point F (which is on the median CF), we can denote:
- FG = 3.5 cm (this is the shorter segment)
- GC = ? (this is the longer segment we want to find)

3. **Using the Ratio**:
Since G divides the median in the ratio of 2:1, we can say that:
- FG : GC = 1 : 2

4. **Setting Up the Equation**:
Let GC = x. According to the ratio, we can express the relationship as:
- FG = 3.5 cm (1 part)
- GC = 2 parts = 2 * FG

5. **Calculating GC**:
Now, we can substitute FG into the equation:
- GC = 2 * 3.5 cm
- GC = 7 cm

6. **Conclusion**:
Therefore, the length of segment GC is 7 cm.

### Final Answer:
GC = 7 cm

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