Question

In ΔABC, ∠ABC = ∠DAC, AB = 8 cm, AC = 4 cm and AD = 5 cm.

1. Prove that ΔACD is similar to ΔBCA.
2. Find BC and CD.
3. Find area of ΔACD : area of ΔABC.

Answer 1

∠ABC = ∠DAC = x  ...(Say)

AB = 8 cm,

AC = 4 cm,

AD = 5 cm.

i. In ΔACD and ΔBCA

∠ABC = ∠DAC   ...(Given)

∠ACD = ∠BCA   ...(Common angles)

`=>` ΔACD ∼ ΔBCA   ...(AA criterion for similarity) (i)

Hence ΔACD is similar to ΔBCA.

ii. In ΔACD and ΔBCA 

ΔACD ∼ ΔBCA   ...[From (i)]

`(AC)/(BC) = (CD)/(CA) = (AD)/(BA)`

`=> (4)/(BC) = (CD)/(4) = (5)/(8)`

`=> (4)/(BC) = (5)/(8)`

`=> BC = (8 xx 4)/(5) = (32)/(5)`

= 6.4 cm

And `(CD)/(4) = (5)/(8)`

`=> CD = (5 xx 4)/(8)`

`=>` CD = 2.5 cm

iii. In ΔACD and ΔBCA 

ΔACD ∼ ΔBCA   ...[From (i)] 

`"Area of ΔACD"/"Area of ΔABC" = ("AC"/"AB")^2`

`=> "Area of ΔACD"/"Area of ΔABC" = 4^2/8^2 = 16/64 = 1/4`