Question

In the given figure, P is a point on AB such that AP : PB = 4 : 3. PQ is parallel to AC.

1. Calculate the ratio PQ : AC, giving reason for your answer.
2. In triangle ARC, ∠ARC = 90° and in triangle PQS, ∠PSQ = 90°. Given QS = 6 cm, calculate the length of AR.

Answer 1

i. Given, AP : PB = 4 : 3.

Since, PQ || AC.

Using Basic Proportionality theorem,

`(AP)/(PB) = (CQ)/(QB)`

`=> (CQ)/(QB) = 4/3`

`=> (BQ)/(BC) = 3/7`  ...(1)

Now, ∠PQB = ∠ACB   ...(Corresponding angles)

∠QPB = ∠CAB   ...(Corresponding angles)

∴ ΔPBQ ~ ΔABC   ...(AA similarity)

`=> (PQ)/(AC) = (BQ)/(BC)`

`=> (PQ)/(AC) = 3/7`    ...[Using (1)]

ii. ∠ARC = ∠QSP = 90°

∠ACR = ∠SPQ     ...(Alternate angles)

∴ ∆ARC ~ ∆QSP   ...(AA similarity)

`=> (AR)/(QS) = (AC)/(PQ)`

`=> (AR)/(QS) = 7/3`

`=> AR = (7 xx 6)/3 = 14  cm`