In `DeltaABC `two medians BE and CF intersect at the point O and P, Q are the midpoints of BO and CO respectively. If the length of PQ = 3cm, then the length of FE will be

Options

A. 3 cm
B. 6 cm
C. 9 cm
D. 12 cm

Answer

To solve the problem, we will use the properties of medians and midpoints in a triangle.

### Step-by-Step Solution:

1. **Understanding the Triangle and Medians**:
- We have triangle \( ABC \) with medians \( BE \) and \( CF \) intersecting at point \( O \).
- Points \( P \) and \( Q \) are the midpoints of segments \( BO \) and \( CO \) respectively.

2. **Given Information**:
- The length of segment \( PQ \) is given as \( 3 \, \text{cm} \).

3. **Using the Midpoint Theorem**:
- According to the midpoint theorem, the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length.
- Since \( P \) and \( Q \) are midpoints of \( BO \) and \( CO \), segment \( PQ \) is parallel to segment \( BC \) and its length is half of \( BC \).

4. **Establishing the Relationship**:
- Therefore, we can write:
\[ PQ = \frac{1}{2} BC \]
- Given \( PQ = 3 \, \text{cm} \), we can substitute this into the equation:
\[ 3 = \frac{1}{2} BC \]

5. **Solving for \( BC \)**:
- To find \( BC \), we multiply both sides by \( 2 \):
\[ BC = 3 \times 2 = 6 \, \text{cm} \]

6. **Finding Length of \( FE \)**:
- Since \( E \) is the midpoint of \( AC \), and using the property of medians, we know:
\[ FE = \frac{1}{2} BC \]
- Substituting the value of \( BC \):
\[ FE = \frac{1}{2} \times 6 = 3 \, \text{cm} \]

### Final Answer:
The length of \( FE \) is \( 3 \, \text{cm} \).

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