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

(i) Prove that ΔGED ∼ ΔGBF.

(ii) Find DE.

(iii) Find DB : AB.

(i) In ΔGED and ΔGBF,

∠DGE = ∠BGF [Vertically opposite angles are equal]

∠GED = ∠GBF [Alternate angles are equal]

∴ ΔGED ∼ ΔGBF (By A.A. axiom)

Hence, proved ΔGED ∼ ΔGBF.

(ii) We know that,

Corresponding sides of similar triangles are proportional.

$\Rightarrow \dfrac{DE}{BF} = \dfrac{EG}{GB} \\[1em] \Rightarrow \dfrac{DE}{30} = \dfrac{8}{16} \\[1em] \Rightarrow \dfrac{DE}{30} = \dfrac{1}{2} \\[1em] \Rightarrow DE = \dfrac{1}{2} \times 30 \\[1em] \Rightarrow DE = 15 \text{ cm}.$

Hence, DE = 15 cm.

(iii) In ΔDBE and ΔABC,

∠EBD = ∠CBA [Common angles]

∠BDE = ∠BAC [Corresponding angles are equal, Since AC ∥ DE]

ΔDBE ∼ ΔABC (By A.A. axiom)

Corresponding sides of similar triangles are proportional.

$\Rightarrow \dfrac{DB}{AB} = \dfrac{DE}{AC} \\[1em] \Rightarrow \dfrac{DB}{AB} = \dfrac{15}{24} \\[1em] \Rightarrow \dfrac{DB}{AB} = \dfrac{5}{8}.$

Hence, DB : AB = 5 : 8.