In the given figure, AC // DE // BF. If AC = 24 cm, EG = 8 cm, GB = 16 cm, BF = 30 cm.

(a) Given,

DE // BF

∴ ∠DEG = ∠GBF (Alternate angles are equal)

∠EDG = ∠GFB (Alternate angles are equal)

⇒ △GED ~ △ GBF (By A.A. axiom)

Hence, proved that △GED ~ △ GBF.

(b) We know that,

Corresponding sides of similar triangle are proportional.

Since, △ GED ~ △ GBF

$$\begin{aligned}\Rightarrow \frac{DE}{BF} = \frac{EG}{BG} \\[1em] \Rightarrow \frac{DE}{30} = \frac{8}{16} \\[1em] \Rightarrow DE = 30 \times \frac{8}{16} \\[1em] \Rightarrow DE = 15 \text{ cm}.\end{aligned}$$

Hence, DE = 15 cm.

(c) In △ BAC and △ BDE,

⇒ ∠BAC = ∠BDE (Corresponding angles are equal)

⇒ ∠ABC = ∠DBE (Common angle)

⇒ △ BAC ~ △ BDE (By A.A. axiom)

We know that,

Corresponding sides of similar triangle are proportional.

Since, △ BAC ~ △ BDE

$$\begin{aligned}\Rightarrow \frac{AB}{BD} = \frac{AC}{DE} \\[1em] \Rightarrow \frac{BD}{AB} = \frac{DE}{AC} \\[1em] \Rightarrow \frac{BD}{AB} = \frac{15}{24} \\[1em] \Rightarrow \frac{BD}{AB} = \frac{5}{8}\end{aligned}$$

⇒ BD : AB = 5 : 8.

Hence, BD : AB = 5 : 8.