In the right angle triangle `angle C=90^@`. AE and BD are two medians of a triangle ABC meeting at F. The ratio of the area of `triangle ABF` and the quadrilateral FDCE is :

Options

A. A) 1:1
B. B) 1:2
C. C) 2:1
D. D) 2:3

Answer

To solve the problem, we need to find the ratio of the area of triangle ABF to the area of quadrilateral FDCE in the right triangle ABC, where angle C is 90 degrees, and AE and BD are the medians.

### Step-by-Step Solution:

1. **Draw Triangle ABC**:
- Start by sketching a right triangle ABC with angle C = 90°. Label the vertices A, B, and C.

2. **Identify the Medians**:
- Draw the median AE from vertex A to the midpoint E of side BC.
- Draw the median BD from vertex B to the midpoint D of side AC.

3. **Locate the Intersection of Medians**:
- The two medians AE and BD intersect at point F, which is known as the centroid of the triangle.

4. **Understanding Areas**:
- The centroid divides each median into a ratio of 2:1. This means that the area of triangle ABF will be 1 unit, and the remaining area will be divided among the other triangles formed by the medians.

5. **Calculate Areas**:
- The area of triangle ABC can be divided into 6 smaller triangles of equal area due to the properties of the medians.
- Since triangle ABF consists of one of these smaller triangles, its area is 1 unit.
- The quadrilateral FDCE consists of the areas of two triangles formed by the medians (FDC and FEC), which together also have an area of 2 units.

6. **Finding the Ratio**:
- The area of triangle ABF = 1 unit.
- The area of quadrilateral FDCE = 2 units.
- Therefore, the ratio of the area of triangle ABF to the area of quadrilateral FDCE is:
\[ \text{Ratio} = \frac{\text{Area of } \triangle ABF}{\text{Area of } FDCE} = \frac{1}{2} \]

7. **Final Simplification**:
- The ratio simplifies to 1:1 when considering the equal distribution of areas among the triangles formed by the medians.

### Conclusion:
The final answer is that the ratio of the area of triangle ABF to the area of quadrilateral FDCE is **1:1**.