Question

Show that the figure formed by joining the midpoints of sides of a rhombus successively is a rectangle.

Answer 1

Let □ABCD be a rhombus. 

P, Q, R and S be the midpoints of sides of □ABCD 

In ΔABC, 

P, Q are the midpoints of AB and BC. 

∴ PQ//AC and PQ = 1/2 AC …………………..(1)

Also in ΔADC, 

S, R are the midpoints of AD and CD. 

∴ SR//AC and SR = 1/2 AC ………………(2)

From (1) and (2); 

PQ // SR and PQ = SR

Similarly QR // PS and QR = PS 

∴ □PQRS is a parallelogram. As the diagonals of a rhombus bisect at right angles. 

∠AOB – 90° 

∴ ∠P = ∠AOB = 90° [opp. angles of //gm PYOX] 

Hence □PQRS is a rectangle as both pairs of opp. sides are equal and parallel, one angle being 90°.