Question

In the given figure, AB and DE are perpendicular to BC.

1. Prove that ΔABC ∼ ΔDEC
2. If AB = 6 cm, DE = 4 cm and AC = 15 cm. Calculate CD.
3. Find the ratio of the area of a ΔABC : area of ΔDEC.

Answer 1

i. Given AB ⊥ BC

DE ⊥ BC

To prove: ΔABC ∼ ΔDEC

Proof: If ΔABC and ΔDEC

∠ABC = ∠DEC = 90°each  ...(Given)

∠C = ∠C   ...(Common)

∴ ΔABC ∼ ΔDEC   ...(AA criterion)

Hence proved.

ii. AB = 6 cm, DE = 4 cm

AC = 15 cm, CD = ?

Since ΔABC ∼ ΔDEC

`=> (AB)/(DE) = (AC)/(CD)`    ...(Corresponding sides of similar Δ's are proportional)

∴ `(6)/(4) = (15)/(CD)`

`=> CD = (15 xx 4)/(6) = 10`.

iii. `"Area of ΔABC"/"Area of ΔDEC" = (AB^2)/(DE^2)`   ...(Area theorem)

= `(36)/(16)`

= `(9)/(4)` or 9 : 4.