Step 1: Understanding the Concept:
In a parallelogram, opposite sides are parallel. Similarity of triangles can be established using alternate interior angles. Step 2: Key Formula or Approach:
From the diagram, \(E\) is a point on \(BC\) and line segment \(DF\) is drawn where \(F\) is on extended side \(AB\).
Wait, looking closely at the figure, \(F\) is on side \(AB\). In \(\triangle FBE\) and \(\triangle FAD\):
Since \(AD \parallel BC\), \(\angle FBE = \angle FAD\) (corresponding angles) and \(\angle FEB = \angle FDA\) (corresponding angles).
Thus, \(\triangle FBE \sim \triangle FAD\). Step 3: Detailed Explanation:
By similarity of \(\triangle FBE \sim \triangle FAD\), the ratios of corresponding sides are equal:
\[ \frac{FB}{FA} = \frac{FE}{FD} \]
Given: \(FB = 3 \text{ cm}\), \(AF = 7 \text{ cm}\) (This is the total side length \(FA\) in the larger triangle), and \(EF = 4 \text{ cm}\).
\[ \frac{3}{7} = \frac{4}{FD} \]
Cross-multiplying:
\[ 3 \times FD = 7 \times 4 \]
\[ 3 \times FD = 28 \]
\[ FD = \frac{28}{3} \text{ cm} \] Step 4: Final Answer:
The length \(FD\) is \(\frac{28}{3} \text{ cm}\).
