ABCD is a rhombus and P, Q, R and S are wthe mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle.

Explanation

Given In quadrilateral ABCD, P, Q, R and S are the mid-points of the sides AB, BC, CD and DA, respectively.
Also, `" "ACbotBD`
To prove PQRS is a rectangle.
Proof Since, `" "ACbotBD`
`therefore" "angleCOD=angleAOD=angleAOB=angleCOB=90^(@)` ltBrgt In `Delta`ADC, S and R are the mid-points of AD and DC respectively, then by mid-point theorem
`" "SR||AC and SR=(1)/(2)AC" "...(i)`

In `Delta`ABC, P and Q are the mid-points of AB and BC respectively, then by mid-point theorem
`" "PQ||AC and PQ=(1)/(2) AC" "...(ii)`
From Eqs. (i) and (ii), `" "PQ||SR and PQ=SR=(1)/(2)AC" "...(iii)`
Similarly, `" "SP||RQ and SP=RQ=(1)/(2)BD" "...(iv)`
Now, in quadrilateral EOFR, `" "OE||FR, OF||ER`
`therefore" " angleEOF=angleERF=90^(@)" "[becauseangleCOD=90^(@)rArrangleEOF=90^(@)]...(v)`
So, PQRS is a rectangle. `" "` Hence proved.