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Question
In the given figure, P is a point on AB such that AP : PB = 4 : 3. PQ is parallel to AC.
- Calculate the ratio PQ : AC, giving reason for your answer.
- In triangle ARC, ∠ARC = 90° and in triangle PQS, ∠PSQ = 90°. Given QS = 6 cm, calculate the length of AR.
Solution
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`
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