In a `Delta ABC, angle B=90^(@)`, median AD and CF intersects at 'O' . Find the ratio of area of `Delta AOC and BDOF`.

Answer

To find the ratio of the areas of triangles AOC and BDOF in triangle ABC where angle B = 90°, follow these steps:

### Step-by-Step Solution:

1. **Draw Triangle ABC**:
- Begin by sketching triangle ABC with angle B as a right angle. Label the vertices as A, B, and C, where B is at the right angle.

2. **Identify the Medians**:
- Draw the median AD from vertex A to the midpoint D of side BC.
- Draw the median CF from vertex C to the midpoint F of side AB.
- These medians intersect at point O.

3. **Understanding the Properties of Medians**:
- Recall that a median divides a triangle into two triangles of equal area. Therefore, median AD divides triangle ABC into two triangles: ABD and ACD, which have equal areas.

4. **Determine Areas of Triangles**:
- Since triangle ABC is divided into two equal areas by median AD, we can denote the area of triangle ABD as \( A \) and the area of triangle ACD as \( A \). Thus, the total area of triangle ABC is \( 2A \).

5. **Area of Triangle AOC**:
- The median CF divides triangle ACD into two smaller triangles: AOC and COF. Since CF is a median, the areas of triangles AOC and COF are equal. Therefore, the area of triangle AOC is \( \frac{A}{2} \).

6. **Area of Quadrilateral BDOF**:
- The area of quadrilateral BDOF can be calculated as the area of triangle ABC minus the area of triangle AOC.
- The area of triangle ABC is \( 2A \) and the area of triangle AOC is \( \frac{A}{2} \). Thus, the area of quadrilateral BDOF is:
\[ \text{Area of BDOF} = \text{Area of ABC} - \text{Area of AOC} = 2A - \frac{A}{2} = \frac{4A}{2} - \frac{A}{2} = \frac{3A}{2} \]

7. **Finding the Ratio of Areas**:
- Now, we can find the ratio of the area of triangle AOC to the area of quadrilateral BDOF:
\[ \text{Ratio} = \frac{\text{Area of AOC}}{\text{Area of BDOF}} = \frac{\frac{A}{2}}{\frac{3A}{2}} = \frac{1}{3} \]

### Final Answer:
The ratio of the area of triangle AOC to the area of quadrilateral BDOF is \( 1:3 \).