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### Given:
- \( D \), \( E \), and \( F \) are midpoints of sides \( AB \), \( BC \), and \( CA \) of \( \triangle ABC \).
- \( \text{Area of } \triangle ABC = 64 \, \text{cm}^2 \).
- We need to find the area of \( \triangle BDE \).

### Concept:
Since \( D \), \( E \), and \( F \) are the midpoints of \( AB \), \( BC \), and \( CA \), the line segments \( DE \), \( EF \), and \( FD \) form a smaller triangle inside the original triangle \( \triangle ABC \).

#### Median Triangle Theorem:
When we connect the midpoints of a triangle, the triangle formed (here \( \triangle DEF \)) will have an area that is exactly one-fourth of the area of the larger triangle \( \triangle ABC \). Therefore, the area of \( \triangle DEF \) will be:
\[
\text{Area of } \triangle DEF = \frac{1}{4} \times 64 \, \text{cm}^2 = 16 \, \text{cm}^2.
\]

However, we are asked to find the area of \( \triangle BDE \), which is half of \( \triangle ABC \) due to the fact that we are only considering part of the triangle formed by two medians and the side \( BD \).

### Solution:
The triangle \( \triangle BDE \) is one-fourth the area of the whole triangle \( \triangle ABC \), because we are dividing it by the medians, and thus it will also have an area of \( 16 \, \text{cm}^2 \).

### Answer:
The area of \( \triangle BDE \) is **16 cm²** (Option C).

Would you like further details on this or have any other questions?

Here are 5 related questions to expand your understanding:
1. How does the area of a median triangle compare to the area of the original triangle?
2. What is the centroid theorem, and how does it relate to midpoints in a triangle?
3. How do you find the area of a triangle given its vertices using coordinates?
4. How would the area change if \( D \), \( E \), and \( F \) were not midpoints?
5. What is the significance of medians in dividing a triangle into equal-area sections?

**Tip**: When solving geometry problems, always remember that median lines and midpoint connections divide a triangle into proportional areas!

D, E, F are midpoints of sides AB, BC and CA of ΔABC, if ar(ΔABC) = 64 cm² then, area of ΔBDE is

Learn how to find the area of ΔBDE when D, E, and F are midpoints of sides AB, BC, and CA of ΔABC. This problem explores key geometric concepts such as the Midpoint Theorem and the Median Triangle Theorem, with an original area of 64 cm². Suitable for students in Grades 9-10.

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