Question

Show that the quadrilateral formed by joining the midpoints of the pairs of adjacent sides of a rhombus is a rectangle.

Answer 1

Consider △ ABC

We know that P and Q are the mid points of AB and BC

By using the midpoint theorem

We know that PQ || AC and PQ = ½ AC

Consider △ ADC

We know that RS || AC and RS = ½ AC

It can be written as PQ || RS and

PR = RS = ½ AC ……. (1)

Consider △ BAD

We know that P and S are the mid points of AB and AD

Based on the midpoint theorem

We know that PS || BD and PS = ½ DB

Consider △ BCD

We know that RQ || BD and RQ = ½ DB

It can be written as PS || RQ and

PS = RQ = ½ DB ……. (2)

By considering equations (1) and (2)

The diagonals intersects at right angles in a rhombus

So we get

∠ EQF = 90^o

We know that RQ || DB

So we get RE || FO

In the same way SR || AC

So we get FR || OE

So we know that OERF is a parallelogram.

We know that the opposite angles are equal in a parallelogram

So we get

∠ FRE = ∠ EOF = 90^o

So we know that PQRS is a parallelogram having ∠ R = 90^o

Therefore, it is proved that the quadrilateral formed by joining the midpoints of the pairs of adjacent sides of a rhombus is a rectangle.