Question

In a ∆ABC, D and E are points on AB and AC respectively such that DE || BC. If AD = 2.4 cm, AE = 3.2 cm, DE = 2 cm and BC = 5 cm, find BD and CE.

Answer 1

It is given that ,, and .

We have to find BD and CE.

Since DE$\parallel$BC, AB is transversal, then

∠ADE = ∠ABC (corresponding angles)

Since DE$\parallel$BC, AC is a transversal, then

∠AED = ∠ACB (corresponding angles)

In ∆ADE and ∆ABC,

∠ADE = ∠ABC (proved above)

∠AED = ∠ACB (proved above)

so, ∆ADE ∼ ∆ABC (Angle Angle similarity)

Since, the corresponding sides of similar triangles are proportional, then

$$\begin{gathered} \frac{\mathrm{AD}}{\mathrm{AB}}=\frac{\mathrm{AE}}{\mathrm{AC}}=\frac{\mathrm{DE}}{\mathrm{BC}} \\ \Rightarrow \frac{\mathrm{AD}}{\mathrm{AB}}=\frac{\mathrm{DE}}{\mathrm{BC}} \\ \Rightarrow \frac{2.4}{2.4+\mathrm{DB}}=\frac{2}{5} \\ \Rightarrow 2.4+\mathrm{DB}=6 \\ \Rightarrow \mathrm{DB}=6-2.4 \\ \Rightarrow \mathrm{DB}=3.6\mathrm{cm} \\ \mathrm{Similarly}, \\ \frac{\mathrm{AE}}{\mathrm{AC}}=\frac{\mathrm{DE}}{\mathrm{BC}} \\ \Rightarrow \frac{3.2}{3.2+\mathrm{EC}}=\frac{2}{5} \\ \Rightarrow 3.2+\mathrm{EC}=8 \\ \Rightarrow \mathrm{EC}=8-3.2 \\ \Rightarrow \mathrm{EC}=4.8\mathrm{cm} \end{gathered}$$

Hence, BD = 3.6 cm and CE = 4.8 cm.