Let's walk through the solution step-by-step.
### Step 1: Calculate the Area of Triangle [tex]\( \triangle ABC \)[/tex]
First, we need to find the area of the triangle [tex]\( \triangle ABC \)[/tex]. Given:
- Length of [tex]\( BC \)[/tex] is 6 cm
- Perpendicular distance from [tex]\( A \)[/tex] to [tex]\( BC \)[/tex] is 8 cm
The formula for the area of a triangle based on its base and height is:
[tex]\[ \text{Area}_{\triangle ABC} = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]
Substituting the given values:
[tex]\[ \text{Area}_{\triangle ABC} = \frac{1}{2} \times 6 \, \text{cm} \times 8 \, \text{cm} \][/tex]
[tex]\[ \text{Area}_{\triangle ABC} = \frac{1}{2} \times 48 \, \text{cm}^2 \][/tex]
[tex]\[ \text{Area}_{\triangle ABC} = 24 \, \text{cm}^2 \][/tex]
### Step 2: Determine the Area of Triangle [tex]\( \triangle DEF \)[/tex]
Triangles [tex]\( \triangle DEF \)[/tex] and [tex]\( \triangle ABC \)[/tex] are similar because [tex]\( D, E, \)[/tex] and [tex]\( F \)[/tex] are the mid-points of [tex]\( BC, CA, \)[/tex] and [tex]\( AB \)[/tex] respectively. The triangle formed by joining the midpoints of the sides of any triangle is always similar to the original triangle, but it has one-fourth the area of the original triangle.
Therefore, the area of [tex]\( \triangle DEF \)[/tex] is:
[tex]\[ \text{Area}_{\triangle DEF} = \frac{1}{4} \times \text{Area}_{\triangle ABC} \][/tex]
Substituting the area of [tex]\( \triangle ABC \)[/tex]:
[tex]\[ \text{Area}_{\triangle DEF} = \frac{1}{4} \times 24 \, \text{cm}^2 \][/tex]
[tex]\[ \text{Area}_{\triangle DEF} = 6 \, \text{cm}^2 \][/tex]
### Conclusion
The area of triangle [tex]\( \triangle DEF \)[/tex] is [tex]\( 6 \, \text{cm}^2 \)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{6 \, \text{cm}^2}
\][/tex]
So, the correct choice is (iii) [tex]\( 6 \, \text{cm}^2 \)[/tex].
