Question

If PQRS is a rectangle where diagonal PR bisects $\angle P$, then which of the following is not correct?

• A. $PQ=QR$
• B. $PR=QS$
• C. $PQ=RP$
• D. $\angle PQS=\angle SQR$

Answer 1

The correct option is C $PQ=RP$



Given that $PQRS$ is a rectangle and PR bisects $\angle P.$
$\therefore \angle SPR=\angle RPQ$ …..(i)
Since $PQRS$ is a rectangle.
$\therefore \angle PQR=\angle PSR=\angle SPQ=\angle SRQ=90°$ …..(ii)
In $\Delta PQR$ and $\Delta PSR$,
$\angle PQR=\angle PSR$ [From (ii)]
$\angle QPR=\angle SPR$ [From (i)]
$PR=PR$ (Common)
$\therefore \Delta PQR\cong \Delta PSR$ (AAS congruence rule)
$\Rightarrow PQ=SP$ (CPCT)
$\Rightarrow PQ=QR=SR=PS$ $(\because PQ=SR$ and $QR=PS)$ …..(iii)
Also, the diagonals of a rectangle are equal.
$\therefore PR=QS$
Now, In $\Delta PQR$
$\angle PQR=90º$
$\Rightarrow PQ<PR$ (PR is the hypotenuse)
$\therefore PQ=PR$ is incorrect.
Hence, the correct answer is option (3).