Question

In the given figure, if points P, Q, R, S are on the sides of parallelogram such that AP = BQ = CR = DS then prove that `square`PQRS is a parallelogram.

Answer 1

Given: `square` ABCD is a parallelogram.

AP = BQ = CR = DS

To prove: `square`PQRS is a parallelogram.

Proof:

AP = CR      ...(Given) ...(i)

`square`ABCD is a parallelogram.

AB = CD      ...(opposite sides of parallelogram)

∴ AP + PB = CR + RD     ...[A-P-B, D-R-C]   ...(ii)

∴ PB = RD       ...[From (i) and (ii)]     ...(iii)

∠ABC ≅ ∠ADC    ...(Opposite angles of parallelogram)

That is, ∠PBQ ≅ ∠RDS      ...(A-P-B, B-Q-C, C-R-D and A-S-D)    ...(iv)

In ΔPBQ and ΔRDS,

Seg PB ≅ Seg RD     ...[From (iii)]

∠PBQ ≅ ∠RDS        ...[From (iv)]

Seg BQ ≅ Seg SD       ...(Given)

∴ ΔPBQ ≅ ΔRDS       ...[SAS test]

Seg PQ ≅ Seg RS       ...(c.s.c.t) ...(v)

Thus, we can prove that, ΔPAS ≅ ΔRCQ,

∴ Seg PS ≅ Seg RQ       ...(vi)

In `square`PQRS,

Seg PQ ≅ Seg RS     ...[From (v)]

Seg PS ≅ Seg RQ      ...[from (vi)]

If the opposite sides of a quadrilateral are congruent, then it is a parallelogram.

∴ `square`PQRS is a parallelogram.