In the given figure, DE ∥ BC and BD = DC.

(i) Prove that DE bisects ∠ADC.

(ii) If AD = 4.5 cm, AE = 3.9 cm and DC = 7.5 cm, find CE.

(iii) Find the ratio AD : DB.

(i) Given, DE ∥ BC.

⇒ ∠ADE = ∠DBC [Corresponding angles are equal] … (1)

⇒ ∠EDC = ∠DCB [Alternate interior angles are equal] … (2)

Given, BD = DC.

⇒ ∠DBC = ∠DCB [Angles opposite to equal sides are equal] … (3)

From (1), (2), and (3), we get:

∠ADE = ∠EDC

Thus, DE bisects ∠ADC.

Hence, DE bisects ∠ADC.

(ii) Given,

AD = 4.5 cm, AE = 3.9 cm, DC = 7.5 cm

Given,

BD = DC = 7.5 cm

By basic proportionality theorem we have,

$\Rightarrow \dfrac{AD}{DB} = \dfrac{AE}{EC} \\[1em] \Rightarrow \dfrac{4.5}{7.5} = \dfrac{3.9}{EC} \\[1em] \Rightarrow 4.5 \times EC = 3.9 \times 7.5 \\[1em] \Rightarrow 4.5 \times EC = 29.25 \\[1em] \Rightarrow EC = \dfrac{29.25}{4.5} \\[1em] \Rightarrow EC = 6.5 \text{ cm.}$

Hence, CE = 6.5 cm.

(iii) By basic proportionality theorem,

$\Rightarrow \dfrac{AD}{DB} = \dfrac{AE}{EC} \\[1em] \Rightarrow \dfrac{AD}{DB} = \dfrac{3.9}{6.5} \\[1em] \Rightarrow \dfrac{AD}{DB} = \dfrac{3}{5}.$

Hence, the ratio AD : DB is 3 : 5.