Question

ABCD is a rectangle Pis the midpoints of AB,Q and R are points in AD and BC respectively such that AQ =BR .Prove that PQ=PR

Answer 1

in the given sum two triangles will form
for them
in ΔAPQ and ΔPRB
AQ=BR (given)
angleQAP=angleRBP (90 rectngle)
AP=BP ( p is midpoint)

so, ΔAPQ similer ΔPBR ( s-a-s)

so, for sure PQ=PR (proved)

Answer 2

In rectangle ABCD
Given that: P is midpoint of AB
therefore, AB = AP + BP
OR, AB = AP + AP --(1)
Also given, AQ = BR --(2)

Now, In tri. AQP and tri. BRP
AP = BP [from(1)]
angle A = angle B (angles of rectangle)
AQ = BR [from (2)]
so,from S. A. S
tri. AQP CONGRUENT tri. BRP
Hence, PQ = PR
by c. p. c. t

proved