Question

In the following circle, QT is a diameter that goes through point S. Find the measure of OP, QR, and QT. Justify your answers?

Answer 1

In the given figure `QT` is the diameter of the circle `OTPQ`.

`arcPQ=arcOQ=>"chord"PQ="chord"OQ`

So in `DeltasQOT and QPT`

`angle QOT=angle QPT="semi circular angle"=90^@`

`OQ=OPand QT " is common hypotenuse"`

So `DeltasQOT and QPT` are congruent

So `PT=OT=12`

Now in `DeltasORT and PRT`

`OT=PT=12`

`angle OTR=angle PTR`,
being circumferential angles on equal arcs OQ and PQ

and `RT` is common

So `DeltasORT and PRT` are congruent by `SAS " rule"`

Hence `PR=OR and angle TRO=angle TRP=90^@`

This gives `OP=2PR=2xx5=10`

No for right `DeltaPRT`

`RT^2=PT^2-PR^2`

`=>RT^2=12^2-5^2=119`

`=>RT=sqrt119`

Again
As in `Delta QPT,angle QPT=90^@` and `PR` is perpedicular on `QT`

So `QRxxRT=PR^2`

`QR=(PR^2)/(RT)=5^2/sqrt119=25/119xxsqrt119`