Question

In the given figure, PQ || AB; CQ = 4.8 cm QB = 3.6 cm and AB = 6.3 cm. Find :

1. `(CP)/(PA)` 
2. PQ
3. If AP = x, then the value of AC in terms of x.

Answer 1

i. In ΔCPQ and ΔCAB,

∠PCQ = ∠ACB  ...(Since PQ || AB, so the angles are corresponding angles)

∠C = ∠C   ...(Common angle)

∴ ΔCPQ ∼ ΔCAB  ...(AA criterion for similarity)

`=> (CP)/(PA) = (CQ)/(QB)`

`=> (CP)/(PA) = (4.8)/(3.6) = (48)/(36) = 4/3`

So, `(CP)/(PA) = 4/3`

ii. In ΔCPQ and ΔCAB,

∠PCQ = ∠ACB ...(Since PQ || AB, so the angles are corresponding angles)

∠C = ∠C ...(Common angle)

∴ ΔCPQ ∼ ΔCAB ...(AA criterion for similarity)

`=> (PQ)/(AB) = (CQ)/(CB)`

`=> (PQ)/(6.3) = (4.8)/(8.4) `

So, PQ = 3.6

iii. In ΔCPQ and ΔCAB,

∠PCQ = ∠ACB  ...(Since PQ || AB, so the angles are corresponding angles)

∠C = ∠C   ...(Common angle)

∴ ΔCPQ ∼ ΔCAB  ...(AA criterion for similarity)

`=> (CP)/(AC) = (CQ)/(CB)`

`=> (CP)/(AC) = (4.8)/(8.4) = 4/7 `

So, if AC is 7 parts, and CP is 4 parts, then PA is 3 parts.

Given, AP = x

or 3 parts = x

`=>` 1 part = `x/3`

`=>` 7 parts = `(7x)/3`

Hence, AC = `(7x)/3`