Fig. 9294BCD is a rhombus and P, Q, R and S are wthe mid-pointsof the

1.	Fig. 9294BCD is a rhombus and P, Q, R and S are wthe mid-pointsof the sides AB, BC CD and DA respectively. Show that the quadrilateral PQRS is arectangle.
Answer» Given: ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. To prove: PQRS is a rectangle.Proof: PS \|\| BD and PS = 1/2 BD [According to midpoint theorem]QR \|\| BD and QR = 1/2 BD [According to midpoint theorem]⇒ PS \|\| QR and PS = QR.PQ \|\| AC and PQ = 1/2 AC [According to midpoint theorem]RS \|\| AC and RS = 1/2 AC [According to midpoint theorem]⇒ PQ \|\| RS and PQ = RS. Opposite sides of the quadrilateral are equal and parallel. so PQRS is a parallelogram.Now , we have to prove that

Fig. 9294BCD is a rhombus and P, Q, R and S are wthe mid-pointsof the sides AB, BC CD and DA respectively. Show that the quadrilateral PQRS is arectangle.

Answer»

Given: ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively.

To prove: PQRS is a rectangle.Proof: PS || BD and PS = 1/2 BD

[According to midpoint theorem]QR || BD and QR = 1/2 BD

[According to midpoint theorem]⇒ PS || QR and PS = QR.PQ || AC and PQ = 1/2 AC

[According to midpoint theorem]RS || AC and RS = 1/2 AC

[According to midpoint theorem]⇒ PQ || RS and PQ = RS.

Opposite sides of the quadrilateral are equal and parallel.

so PQRS is a parallelogram.Now , we have to prove that