Question

In the given figure, \(AB \parallel DE\) and \(AC \parallel DF\). Show that \(\triangle ABC \sim \triangle DEF\). If \(BC = 10\) cm, \(EB = CF = 5\) cm and \(AB = 7\) cm, then find the length DE.

Hint:

When proving similarity with parallel lines, look for the 'F' shape for corresponding angles. Always ensure you add up the segments correctly to find the full length of the side of the larger triangle.

Answer 1

Step 1: Understanding the Concept:
Parallel lines create equal corresponding angles when intersected by a transversal. This allows us to prove triangle similarity via the AA (Angle-Angle) criterion.
Step 2: Detailed Explanation:
Part 1: Showing Similarity
In \(\triangle ABC\) and \(\triangle DEF\):
1. Since \(AB \parallel DE\) and \(EF\) is a transversal, \(\angle ABC = \angle DEF\) (Corresponding angles).
2. Since \(AC \parallel DF\) and \(EF\) is a transversal, \(\angle ACB = \angle DFE\) (Corresponding angles).
Therefore, \(\triangle ABC \sim \triangle DEF\) by the AA Similarity Criterion.
Part 2: Finding Length DE
Since the triangles are similar, their corresponding sides are proportional:
\[ \frac{AB}{DE} = \frac{BC}{EF} \]
Given: \(BC = 10\) cm, \(EB = 5\) cm, \(CF = 5\) cm.
From the figure, \(EF = EB + BC + CF\).
\[ EF = 5 + 10 + 5 = 20 \text{ cm} \]
Now, substitute the values into the proportionality equation:
\[ \frac{7}{DE} = \frac{10}{20} \]
\[ \frac{7}{DE} = \frac{1}{2} \]
\[ DE = 7 \times 2 = 14 \text{ cm} \]
Step 3: Final Answer:
The length of \(DE\) is 14 cm.