In the adjoining figure, ∠ADE = ∠ABC, AE = 8 cm, EB = 7 cm, BC = 9 cm, AD = 10 cm and DC = 2 cm. Then the length of DE is:

1. 6 cm

2. 6.75 cm

3. 7.8 cm

4. 13.5 cm

From figure,

AB = AE + EB = 8 + 7 = 15 cm

AC = AD + DC = 10 + 2 = 12 cm

In ΔABC and ΔADE

∠BAC = ∠DAE [Common angle]

∠ABC = ∠ADE [Given]

∴ ΔABC ∼ ΔADE (By A.A. axiom).

We know that,

In similar triangles the ratio of corresponding sides are equal.

$\Rightarrow \dfrac{AD}{AB} = \dfrac{AE}{AC} = \dfrac{DE}{BC} \\[1em] \text{Let's Consider} \\[1em] \Rightarrow \dfrac{AD}{AB} = \dfrac{DE}{BC} \\[1em] \Rightarrow \dfrac{10}{15} = \dfrac{DE}{9} \\[1em] \Rightarrow DE = \dfrac{10 \times 9}{15} \\[1em] \Rightarrow DE = \dfrac{90}{15} \\[1em] \Rightarrow DE = 6 \text{ cm.}$

Hence, option 1 is the correct option.