Question

In the given figure, $OA=OB$ then which of the following is correct ?

• A. $PT=ST$
• B. $PR=SQ$
• C. $PT+RT+PR=ST+QT+SQ$
• D. $PQ=RS$

Answer 1

Answer: (A), (B), (C), (D)

$PQ=RS$


As $O$ is the centre and $OA$ and $OB$ are perpendicular to $PQ$ and $RS$ respectively so $PQ=RS$ as equal chords are equidistance from the centre of the circle.

We can see that point R and Q both are in the same minor segment PS so, major arc PS will subtend equal angles at point R and Q.
Hence, $\angle PRS=\angle PQS$

We can also see that point P and S both are in the same major segment QR so, minor arc QR will subtend equal angles at point P and S.
Hence, $\angle QPR=\angle RSQ$


Now, in $△PRT$ and $△SQT$
$\angle RPT=\angle QST\text{(angle made on same major segment QR of circle by minor arc QR)}$

$\angle PTR=\angle STQ\text{(Vertically opposite angle)}$

$\angle PRT=\angle TQS\text{(angle made on same minor segment PS of circle by major arc PS)}$

$△PRT\cong △SQT$ (AAS) $PT=ST$ (Corresponding pair of congruent triangle) $RT=QT$(Corresponding pair of congruent triangle) $PR=SQ$(Corresponding pair of congruent triangle)

$\therefore PT+RT+PR=ST+QT+SQ$

Hence, all options are correct.