Question

In the given figure, $AB \parallel EF$. If $AB = 24\text{ cm}$, $EF = 36\text{ cm}$ and $DA = 7\text{ cm}$, then $AE$ equals

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

Hint:

Always simplify the ratio of parallel sides ($\frac{24}{36} = \frac{2}{3}$) first before performing algebraic cross-multiplication.
This keeps the numerical calculations simple and error-free!

Answer 1

The Correct Option is C

Step 1: Identify the similar triangles by the basic proportionality idea.
Since $AB \parallel EF$, triangles $DAB$ and $DEF$ are similar, so corresponding sides along the two transversals are in the same ratio.
Step 2: Write the ratio using $DE$ directly instead of $AE$.
\[ \frac{DA}{DE} = \frac{AB}{EF} = \frac{24}{36} = \frac{2}{3} \]
Step 3: Solve for $DE$ first, then subtract.
Since $DA = 7$, we get $DE = \frac{3}{2} \times 7 = 10.5\text{ cm}$. Then
\[ AE = DE - DA = 10.5 - 7 = 3.5\text{ cm} \]
Step 4: Conclude.
This matches option (3).
\[ \boxed{3.5\text{ cm}} \]