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

To find the length of \(AE\), we use the property of similar triangles. Since \(AB \parallel EF\), triangles \(DAB\) and \(DEF\) are similar by the Basic Proportionality Theorem (or Thales' theorem).

According to the theorem:

\(\frac{DA}{DE} = \frac{AB}{EF}\)

Given:

• \(AB = 24 \, \text{cm}\)
• \(EF = 36 \, \text{cm}\)
• \(DA = 7 \, \text{cm}\)

Let's denote \(DE = DA + AE\). We need to find \(AE\), so:

\(\frac{7}{7 + AE} = \frac{24}{36}\)

Simplifying the ratio on the right-hand side:

\(\frac{24}{36} = \frac{2}{3}\)

Now, equate the ratios:

\(\frac{7}{7 + AE} = \frac{2}{3}\)

Cross-multiply to solve for \(AE\):

\(3 \times 7 = 2 \times (7 + AE)\)

\(21 = 14 + 2 \times AE\)

\(21 - 14 = 2 \times AE\)

\(7 = 2 \times AE\)

\(AE = \frac{7}{2} = 3.5 \, \text{cm}\)

Thus, the length of \(AE\) is 3.5 cm.