Question

In the adjoining figure, DE || BC and `(DE)/(BC) = 1/3`. If AD = 1.5 cm, then find the length of BD.

Answer 1

1. Identify similar triangles

In ΔABC, it is given that DE || BC. Because these lines are parallel, the corresponding angles are equal:

∠ADE = ∠ABC

∠AED = ∠ACB

∠A is common to both triangles.

Therefore, ΔADE ∼ ΔABC by the AA (Angle-Angle) similarity criterion.

2. Use the ratio of corresponding sides

Since the triangles are similar, the ratios of their corresponding sides are equal:

`(AD)/(AB) = (DE)/(BC)`

We are given that `(DE)/(BC) = 1/3`, so:

`(AD)/(AB) = 1/3`

3. Calculate the length of AB

Substitute the known value of AD = 1.5 cm into the equation:

`1.5/(AB) = 1/3`

Cross-multiply to solve for AB:

AB = 1.5 × 3

AB = 4.5 cm

4. Find the length of BD

The side AB is the sum of segments AD and BD:

AB = AD + BD

4.5 = 1.5 + BD

Subtract 1.5 from both sides:

BD = 4.5 – 1.5

BD = 3 cm

The length of BD is 3 cm.