In `Delta ` ABC, D, E, F are mid-points of AB, BC, CA respectively and `angle B = 90^@`, AB = 6 cm, BC = 8 cm. Then area of `Delta` DEF (in sq.cm) is

Options

A. 24
B. 28
C. 6
D. 12

Answer

To find the area of triangle DEF, we will follow these steps:

### Step 1: Identify the dimensions of triangle ABC
In triangle ABC, we know:
- Angle B = 90°
- AB = 6 cm
- BC = 8 cm

### Step 2: Use the midpoint theorem
D, E, and F are the midpoints of sides AB, BC, and CA respectively. According to the midpoint theorem, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

### Step 3: Calculate the lengths of segments DF and DE
- Since D is the midpoint of AB, the length AD = DB = AB/2 = 6 cm / 2 = 3 cm.
- Since F is the midpoint of AC, we need to find the length of AC. We can use the Pythagorean theorem:
\[ AC = \sqrt{AB^2 + BC^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ cm} \]
- Therefore, AF = FC = AC/2 = 10 cm / 2 = 5 cm.
- Since E is the midpoint of BC, the length BE = EC = BC/2 = 8 cm / 2 = 4 cm.

### Step 4: Calculate the lengths of DF and DE
- Using the midpoint theorem:
- DF is parallel to BC and DF = BC/2 = 8 cm / 2 = 4 cm.
- DE is parallel to AC and DE = AC/2 = 10 cm / 2 = 5 cm.

### Step 5: Calculate the area of triangle DEF
Triangle DEF is a right triangle where:
- Base DF = 4 cm
- Height DE = 3 cm (since D and E are midpoints, DE is vertical to DF)

Using the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \]
Substituting the values:
\[ \text{Area} = \frac{1}{2} \times 4 \text{ cm} \times 3 \text{ cm} = \frac{12}{2} = 6 \text{ cm}^2 \]

### Final Answer
The area of triangle DEF is **6 cm²**.

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