Question

Given below are some triangles and lengths of line segments. Identify in which figures, ray PM is the bisector of `/_QPR`.

Answer 1

`PQ=7,PR=3,QM=3.5` and `MR=1.5`………..(Given)
`(PQ)/(PR=7/3`………1
`(QM)/(MR)=3.5/1.5=(3.5xx10)/(1.5xx10)=35/15=7/3`….2
In `DeltaQPR`,
`(PQ)/(PR)=(QM)/(MR)`…….[From 1 and 2]
`:.` by converse of angle bisector theorem,
ray PM bisects `/_QPR`.
2. `PR=7,PQ=10,RM=6` and `MQ=8` ........(Given)
`(PR)/(PQ)=7/10`.......1
`(RM)/(MQ)=6/8=3/4`..........2
`:.` from 1 and 2
`(PR)/(PQ)!=(RM)/(MQ)`
`:.`ray PM is not the bisector of `/_RPQ`
3. `PQ9,PR=10, QM=3.6` and `MR=4`.(Given)
`(PQ)/(PR)=9/10`........1
`(QM)/(MR)=3.6/4=(3.6xx10)/(4xx10)=36/40=9/10`.......2
In `DeltaQPR`,
`(PQ)/(PR)=(QM)/(MR)`.........[From 1 and 2]
`:.` By converse of angle bisector theorem.
ray PM bisects `/_QPR`.