O is centroid of `DeltaABC` and AD, BE, CF are its three medians. If area of `DeltaAOE` is 15 `cm^(2)`, then area of quadrilateral BDOF is -

Options

A. 20 `cm^(2)`
B. 30 `cm^(2)`
C. 40 `cm^(2)`
D. 25 `cm^(2)`

Answer

To solve the problem, we need to find the area of quadrilateral BDOF given that the area of triangle AOE is 15 cm² and O is the centroid of triangle ABC.

### Step-by-Step Solution:

1. **Identify the Medians and Centroid**:
- In triangle ABC, the medians AD, BE, and CF intersect at point O, which is the centroid of the triangle. The centroid divides each median in the ratio 2:1.

2. **Area of Triangle AOE**:
- We are given that the area of triangle AOE is 15 cm².

3. **Understanding the Areas of the Other Triangles**:
- Since O is the centroid, it divides triangle ABC into six smaller triangles of equal area. Therefore, the areas of triangles AOE, BOE, COE, AOD, BOD, and COF are all equal.
- If area AOE = 15 cm², then the areas of triangles BOE, COE, AOD, BOD, and COF are also 15 cm² each.

4. **Calculate the Area of Quadrilateral BDOF**:
- Quadrilateral BDOF consists of triangles BOD and BOF.
- The area of triangle BOD = 15 cm² (as established).
- The area of triangle BOF = 15 cm² (as established).
- Therefore, the area of quadrilateral BDOF = Area of triangle BOD + Area of triangle BOF = 15 cm² + 15 cm² = 30 cm².

5. **Conclusion**:
- The area of quadrilateral BDOF is 30 cm².

### Final Answer:
The area of quadrilateral BDOF is **30 cm²**.