Question

PQRS is a kite in which PQ=PS and QR=SR.Show that PR is the perpendicular bisector of diagonal QS.

Answer 1

ΔPQR and ΔPSR 

PQ = PS, QS = SR, PR = RP 
ΔPQR = ΔPSR (by SSS) 

∠1 = ∠2 (CPCT) 

ΔPOQ and ΔPOS 

∠1 = ∠2 

PQ = PS 

OP = OP 

ΔPOQ = ΔPOS (By SAS) 
∠3 = ∠4 (CPCT) 

∠3 + ∠4 = 180 

∠3 = ∠4 = 90° 

ΔPOQ and ΔSOR 

∠1 = ∠5 (alternate interior angle) 

PQ = RS 

∠6 = ∠7 (alternate Interior angle) 

ΔPOQ = ΔSOR (by ASA) 

PO = OR and OQ = OS (by CPCT) 

So, PR  is perpendicular bisector of QS

Answer 2

ΔPQR and ΔPSR

PQ = PS, QS = SR, PR = RP

ΔPQR = ΔPSR (by SSS)

∠1 = ∠2 (CPCT)

ΔPOQ and ΔPOS

∠1 = ∠2

PQ = PS

OP = OP

ΔPOQ = ΔPOS (By SAS)

∠3 = ∠4 (CPCT)

∠3 + ∠4 = 180

∠3 = ∠4 = 90°

ΔPOQ and ΔSOR

∠1 = ∠5 (alternate interior angle)

PQ = RS

∠6 = ∠7 (alternate Interior angle)

ΔPOQ = ΔSOR (by ASA)

PO = OR and OQ = OS (by CPCT)

So, PR is perpendicular bisector of QS