The figure obtained by joining the mid-points of the sides of a rhombus, taken in order, is

Options

A. a rhombus
B. a rectangle
C. a square
D. any parallelogram

Answer

B

Explanation

Let ABCDn be a rhombus in which P, Q , R and S are the mid-points of sides AB, BC, CD and DA, respectively.

Join AC, PR and SQ
In `Delta`ABC, P is the mid-point of AB and Q is the mid-point of BC.
`rArr" "PQ||AC andPQ=(1)/(2)AC" "`[by using mid-point theorem]...(i)
Similarly, in `Delta`DAC,
`SR||AC and SR=(1)/(2)AC" "...(ii)`
From Eqs. (i) and (ii),
`" "PQ||SR and PQ=SR`
So, PQRS is a parallelogram. ltBrgt Also, ABQS is a parallelogram. ltBrgt AB=SQ`" "...(iii)`
`" "` [opposite sides of a parallelogram are equal]
Similarly, PBCR is a parallelogram. ltBrgt `rArr" "BC=PR" "`[oppsite sides of a parallelogram are equal]
`rArr" "AB=PR" "[becauseBC=AB` sides of a rhombus]
`rArr" "SQ=PR" "`[from Eq. (iii)]
So, the diagonals of a parallelogram are equal.
Hence, PQRS is a rectangle.