Question

In the given figure, \(AB \parallel EF\). If \(AB = 24\) cm, \(EF = 36\) cm and \(DA = 7\) cm, then \(AE\) equals

• A. \(2.5\) cm
• B. \(10.5\) cm
• C. \(3.5\) cm
• D. \(\frac{14}{3}\) cm

Hint:

Similar triangles often appear in "ladder" or "parallel line" diagrams. Identify the common vertex (D) to set up the correct side ratio.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Since \(AB \parallel EF\), the triangles \(\triangle DAB\) and \(\triangle DEF\) are similar by AA similarity criterion (\(\angle D\) is common and corresponding angles are equal).
Step 2: Key Formula or Approach:
For similar triangles, the ratio of corresponding sides is equal:
\[ \frac{DA}{DE} = \frac{AB}{EF} \]
Step 3: Detailed Explanation:
Given: \(AB = 24\), \(EF = 36\), \(DA = 7\).
Let \(AE = x\). Then \(DE = DA + AE = 7 + x\).
\[ \frac{7}{7 + x} = \frac{24}{36} \]
Simplify the fraction:
\[ \frac{7}{7 + x} = \frac{2}{3} \]
Cross-multiply:
\[ 21 = 2(7 + x) \]
\[ 21 = 14 + 2x \]
\[ 2x = 21 - 14 = 7 \]
\[ x = \frac{7}{2} = 3.5 \text{ cm} \]
Step 4: Final Answer:
The length of \(AE\) is \(3.5\) cm.