Question

In the given figure, DE $\parallel$ BC. If AD : AB = 1 : 3 and AE = 2.5 cm, then AC equals

• A. 7.5 cm
• B. 5 cm
• C. 10 cm
• D. 2.5 cm

Hint:

Always look at the given ratio carefully.
Sometimes the question gives \(AD : DB\) instead of \(AD : AB\).
Since it is given as \(AD : AB = 1 : 3\), the total side \(AC\) is simply 3 times the smaller part \(AE\).
Calculating \(3 \times 2.5 = 7.5\text{ cm}\) takes only a few seconds!

Answer 1

The Correct Option is A

Step 1: State the given information.
In triangle ABC, DE is parallel to BC (DE \(\parallel\) BC). We are given AD : AB = 1 : 3 and AE = 2.5 cm. We need to find AC.
Step 2: Apply the Basic Proportionality Theorem (Thales Theorem).
Since DE \(\parallel\) BC, by BPT: \(\frac{AD}{AB} = \frac{AE}{AC}\).
Step 3: Substitute the given values.
\(\frac{1}{3} = \frac{2.5}{AC}\).
Step 4: Solve for AC.
\(AC = 2.5 \times 3 = 7.5 \text{ cm}\).
Step 5: Verify using similar triangles.
Triangles ADE and ABC are similar (AA criterion), so corresponding sides are proportional: \(\frac{AD}{AB} = \frac{AE}{AC} = \frac{1}{3}\). This confirms \(AC = 3 \times AE = 7.5 \text{ cm}\).
Step 6: Select the correct option.
\(AC = 7.5 \text{ cm}\), which is option 1.
\[ \boxed{7.5 \text{ cm}} \]