If D,E and F are respectively the midpoint of the sides BC,CA and AB of a `Delta ABC` and the area of `Delta ABC = 36 cm^2` , then area of `Delta DEF` is -----

Options

A. `12 cm^2`
B. `9 cm^2`
C. `18 cm^2`
D. `8 cm^2`

Answer

To find the area of triangle DEF, which is formed by the midpoints of the sides of triangle ABC, we can use the properties of similar triangles and the fact that the area of a triangle formed by the midpoints is a quarter of the area of the original triangle.

### Step-by-Step Solution:

1. **Identify the Midpoints**:
Let D, E, and F be the midpoints of sides BC, CA, and AB of triangle ABC, respectively.

2. **Understand the Relationship**:
According to the Midpoint Theorem, triangle DEF is similar to triangle ABC. The sides of triangle DEF are half the length of the corresponding sides of triangle ABC.

3. **Calculate the Area Ratio**:
The area of similar triangles is proportional to the square of the ratio of their corresponding sides. Since the sides of triangle DEF are half the length of the sides of triangle ABC, the ratio of the sides is 1:2.

4. **Area Ratio Calculation**:
The area ratio will be:
\[ \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

5. **Find the Area of Triangle DEF**:
Given that the area of triangle ABC is 36 cm², we can find the area of triangle DEF by multiplying the area of triangle ABC by the area ratio:
\[ \text{Area of } \triangle DEF = \text{Area of } \triangle ABC \times \frac{1}{4} = 36 \times \frac{1}{4} = 9 \text{ cm}^2 \]

### Final Answer:
The area of triangle DEF is **9 cm²**.