Question

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

1. ${\displaystyle \frac{CP}{PA}}$ 
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)

${\displaystyle \Rightarrow \frac{CP}{PA}=\frac{CQ}{QB}}$

${\displaystyle \Rightarrow \frac{CP}{PA}=\frac{4.8}{3.6}=\frac{48}{36}=\frac{4}{3}}$

So, ${\displaystyle \frac{CP}{PA}=\frac{4}{3}}$

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)

${\displaystyle \Rightarrow \frac{PQ}{AB}=\frac{CQ}{CB}}$

${\displaystyle \Rightarrow \frac{PQ}{6.3}=\frac{4.8}{8.4}}$

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)

${\displaystyle \Rightarrow \frac{CP}{AC}=\frac{CQ}{CB}}$

${\displaystyle \Rightarrow \frac{CP}{AC}=\frac{4.8}{8.4}=\frac{4}{7}}$

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

Given, AP = x

or 3 parts = x

${\displaystyle \Rightarrow }$ 1 part = ${\displaystyle \frac{x}{3}}$

${\displaystyle \Rightarrow }$ 7 parts = ${\displaystyle \frac{7x}{3}}$

Hence, AC = ${\displaystyle \frac{7x}{3}}$