In a △ABC, D and E are points on the sides 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.

The below figure shows △ABC and the points D and E on the sides AB and AC respectively:

Considering △ABC and △ADE,

∠A = ∠A (Common angles)

∠ADE = ∠ABC (Corresponding angles are equal)

Hence by AA axiom △ABC ~ △ADE. Since triangles are similar so the ratio of the corresponding sides are equal,

$\therefore \dfrac{AD}{AB} = \dfrac{AE}{AC} = \dfrac{DE}{BC}$

Consider $\dfrac{AD}{AB} = \dfrac{DE}{BC}$

$\Rightarrow \dfrac{2.4}{AB} = \dfrac{2}{5} \\[1em] \Rightarrow AB = \dfrac{2.4 \times 5}{2} \\[1em] \Rightarrow AB = \dfrac{12}{2} \\[1em] \Rightarrow AB = 6.$

Now consider $\dfrac{AE}{AC} = \dfrac{DE}{BC}$

$\Rightarrow \dfrac{3.2}{AC} = \dfrac{2}{5} \\[1em] \Rightarrow AC = \dfrac{3.2 \times 5}{2} \\[1em] \Rightarrow AC = \dfrac{16}{2} \\[1em] \Rightarrow AC = 8.$

From figure we see that,

⇒ BD = AB - AD = 6 - 2.4 = 3.6 cm.

⇒ CE = AC - AE = 8 - 3.2 = 4.8 cm.

Hence, the length of BD = 3.6 cm and CE = 4.8 cm.