For a triangle `ABC, D, E, F` are the mid-points of its sides. If `DeltaABC = 24` sq. units then `DeltaDEF `is

Options

A. 4 sq. units
B. 6 sq. units
C. 8 sq. units
D. 12 sq. units

Answer

To find the area of triangle DEF, which is formed by connecting the midpoints of triangle ABC, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information**:
We know that the area of triangle ABC is given as 24 square units.

2. **Understanding the Midpoints**:
Let D, E, and F be the midpoints of sides BC, CA, and AB respectively. By connecting these midpoints, we form triangle DEF.

3. **Applying the Midpoint Theorem**:
According to the Midpoint Theorem, when the midpoints of two sides of a triangle are connected, the line segment connecting these midpoints is parallel to the third side and half its length. Thus, triangle DEF is similar to triangle ABC.

4. **Finding the Ratio of Areas**:
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Since each side of triangle DEF is half the length of the corresponding side of triangle ABC, the ratio of the sides is:
\[ \text{Ratio of sides} = \frac{1}{2} \]
Therefore, the ratio of the areas is:
\[ \text{Ratio of areas} = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

5. **Calculating the Area of Triangle DEF**:
Since the area of triangle ABC is 24 square units, we can find the area of triangle DEF as follows:
\[ \text{Area of } \triangle DEF = \frac{1}{4} \times \text{Area of } \triangle ABC = \frac{1}{4} \times 24 = 6 \text{ square units} \]

Thus, the area of triangle DEF is **6 square units**.