Question

5. * In a parallogram ABCD, P divides AB in the ratio 2 : 5 and Q divides DC in the ratio 3 : 2. If AC and PQ intersect at R, find the ratios AR : RC and PR : RQ

Answer 1

Answer

AR: RC = 1 : 1 and PR : RQ = 1 : 1

Step-by-step explanation:

Using figure 1 as shown below.

$In\triangle ARP and \triangle QRC.$

$\angle ARP =\angle QRC (vertically\ opposite\ angles)$

$\angle PAR =\angle QCR \ (\because AB \parallel CD)$

$\therefore \triangle ARP\sim \triangle QRC$$(by AA similarity)$

$\therefore\frac{AP}{AR} =\frac{QC}{RC}$

$\frac{2x}{AR} =\frac{2x}{RC}$

$AR: RC = 1: 1$

In the similar way,

$\frac{QC}{RQ} =\frac{AP}{PR}$

$\frac{PR}{RQ} = \frac{AP}{QC} =\frac{2x}{2x}$

$Therefore, PR: RQ = 1: 1$