Let's analyze and solve this problem step by step.

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!