Question

11. Given: TQ bisects ZRTP, QS || PT
RT = 30. Find QP, RS and QS

Answer 1

QP=24 cm

RS=11.25 cm

QS=18.75 cm

Step-by-step explanation:

Given that TQ bisects <RTP

`therefore <QTP=<QTR\\`(1)

consider ΔRQS and ΔRPT

QS||PT,RP and RT are transversals

`<SQT=<QTP`(alternate angles)(2)

comparing (1) and (2)

`<SQT=<STQ` and triangle SQT is isocelus

Therefore SQ=ST(sides opposite to equal angles in an isocelus triangle)

Therefore <RQS=<RPT(corresponding angles)

<RSQ=<RTP(corresponding angles)

therefore by AA criterion for similarity

ΔRQS~ΔRPT

According to the property of similar triangles

`RQ/RP=RS/RT=QS/PT`
`9/RP=X/30=QS/50\\9/RP=X/30=(30-X)/50\\X/30=(30-X)/50\\50X=30(30-X)\\50X=900-30X\\50X+30X=900\\80X=900\\X=900/80=11.25\\RS=11.25 cm\\QS=30-X=30-11.25\\QS=18.75 cm\\9/RP=X/30\\9/RP=11.25/30\\9*30/11.25=RP\\RP=24 cm`