Question

In the Adjoining figure DE \(||\) BC and \(\frac{\text{AD}}{\text{DB}} = \frac{3}{5}\) if AC = 4.8 cm. Then AE equals :

• A. 1.8
• B. 2.8
• C. 1.9
• D. 2.9

Hint:

1. Given DE \(||\) BC and \(\frac{\text{AD}}{\text{DB}} = \frac{3}{5}\). 2. This means AD is 3 parts and DB is 5 parts. So, AB is \(3+5 = 8\) parts. 3. The ratio \(\frac{\text{AD}}{\text{AB}} = \frac{3}{8}\). 4. By Basic Proportionality Theorem (or its corollary for similar triangles \(\triangle ADE \sim \triangle ABC\)): \(\frac{\text{AD}}{\text{AB}} = \frac{\text{AE}}{\text{AC}}\). 5. Substitute known values: \(\frac{3}{8} = \frac{\text{AE}}{4.8}\). 6. Solve for AE: AE = \(\frac{3 \times 4.8}{8} = \frac{14.4}{8} = 1.8\) cm.

Answer 1

The Correct Option is A

Concept: This problem uses the Basic Proportionality Theorem (BPT), also known as Thales' Theorem, and its corollaries. If a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then the other two sides are divided in the same ratio. A corollary is that if DE \(||\) BC in \(\triangle \text{ABC}\), then \(\frac{\text{AD}}{\text{AB}} = \frac{\text{AE}}{\text{AC}}\). Step 1: Understand the given ratio and relate it to sides of \(\triangle \text{ABC}\) Given: DE \(||\) BC in \(\triangle \text{ABC}\). Given ratio: \(\frac{\text{AD}}{\text{DB}} = \frac{3}{5}\). This means we can let AD = \(3k\) and DB = \(5k\) for some constant \(k\). Then, the full side AB = AD + DB = \(3k + 5k = 8k\). So, the ratio \(\frac{\text{AD}}{\text{AB}} = \frac{3k}{8k} = \frac{3}{8}\). Step 2: Apply the corollary of the Basic Proportionality Theorem Since DE \(||\) BC, by the corollary of BPT: \[ \frac{\text{AD}}{\text{AB}} = \frac{\text{AE}}{\text{AC}} \] Step 3: Substitute the known values into the proportion We found \(\frac{\text{AD}}{\text{AB}} = \frac{3}{8}\). Given AC = 4.8 cm. Let AE = \(x\) cm (this is what we need to find). Substituting these into the proportion: \[ \frac{3}{8} = \frac{\text{AE}}{4.8} \] \[ \frac{3}{8} = \frac{x}{4.8} \] Step 4: Solve for AE (\(x\)) To solve for \(x\), multiply both sides by 4.8: \[ x = \frac{3}{8} \times 4.8 \] \[ x = \frac{3 \times 4.8}{8} \] We can simplify \(4.8/8\). Note that \(48/8 = 6\), so \(4.8/8 = 0.6\). \[ x = 3 \times 0.6 \] \[ x = 1.8 \] So, AE = 1.8 cm. This matches option (1).