P, Q, R and S are respectively the mid-points of sides AB, BC, CD and DA of quadrilateral ABCD in which AC=BD and `ACbotBD`. Prove that PQRS is a square.

Explanation

Given In quadrilateral ABCD, P, Q, R and S are the mid-points of the sides AB, BC, CD and DA, respectively.
Also, `" "AC=BD and ACbotBD`.
To prove PQRSk is a square.
Proof Now, in `Delta`ADC, S and R are the mid points of the sides 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, 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, in `Delta`ABD, by mid-point theorem,
`" "SP||BD and SP=(1)/(2)BD=(1)/(2)AC " "`[given, AC=BD]...(iv)
and `Delta`BCD, by mid-point theorem,
`" "RQ||BD and RQ = (1)/(2)BD=(1)/(2)AC" "`[given, BD=AC]...(v)
From Eqs. (iv) and (v),
`" "SP=RQ=(1)/(2)AC" "...(vi)`
From Eqs. (iii) and (vi), ltBrgt `" "PQ=SR=SP=RQ`
Thus, all four sides are equal.
Now, in quadrilateral OERF, `" "OE||FR and OF||ER`
`therefore" "angleEOF=angleERF=90^(@)`
`" "[becauseAC bot DBrArr=angleDOC=angleEOF=90^(@)` as opposite angles of a parallelogram]
`therefore " "angleQRS=90^(@)`
Similarly, `" "angleRQS=90^(@)`
So, PQRS is a square. `" "` Hence proved.