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

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

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(AP)/(AR) =(QC)/(RC)`

`(2x)/(AR) =(2x)/(RC)`

`AR: RC = 1: 1`

In the similar way,

`(QC)/(RQ) =(AP)/(PR)`

`(PR)/(RQ) = (AP)/(QC) =(2x)/(2x)`

`Therefore, PR: RQ = 1: 1`