In the given figure, ∠ADE = ∠B

i. Show that ΔABC ~ ΔADE

ii. If AD = 3.8cm, AE = 3.6 cm, BE = 2.1cm, BC = 4.2cm. find DE.

(i) Given, ∠ADE = ∠B = θ,∠C = 90°

⇒ ∠A = 90° -θ

⇒ if ∠A = 90°- θ, ∠B = θ

⇒ ∠ AED = 90°

⇒ now, comparing Δ ABC with Δ AED we have

∠ A common in both triangles

∠ C = ∠ AED = 90°

∠ ADE = ∠ B

∴ By AAA property we have in two triangles if the angles are equal, then sides opposite to the equal angles are in the same ratio (or proportional) and hence the triangles are similar.

Hence, Δ ABC and Δ ADE are similar triangles.

(ii) Given, Ad = 3.8cm, AE = 3.6cm, BE = 2.1cm, BC = 4.2cm

Need to find DE.

As Δ ABC and Δ ADE are similar triangles we have

⇒

⇒ AE + BE = 3.6 + 2.1 = 5.7

⇒

⇒ DE = 4.2 ×

= 2.8cm

Hence, the value of DE is 2.8cm