Show that the points P(2,1), Q(-1,3) , R (-5,-3) and
S (-2,-5) are the vertices of a square .

Explanation

`P(2,1),Q(-1,3),R(-5,-3)` and `S(-2,-5)`
By distance formula
`PQ=sqrt((1-2)^(2)+(3-1)^(2))`
`=sqrt((-3)^(2)+(2)^(2))`
`=sqrt(9+4)`
`=sqrt(13)`………….1
`QR=sqrt([-5-(-1)]^(2)+(-3-3)^(2))`
`=sqrt((-4)^(2)+(-6)^(2))`
`=sqrt(16+36)`
`=sqrt(52)`
`=sqrt(2xx2xx13)`
`=2sqrt(13)`..............2
`RS=sqrt([-2-(-5)]^(2)+[-5-(-3)]^(2))`
`=sqrt((-2+5)^(2)+(-5+3)^(2))`
`=sqrt(3^(2)+(-2)^(2))`
`=sqrt(9+4)`
`=sqrt(13)` ..............3
`PS=sqrt((-2-2)^(2)+(-5-1)^(2))`
`=sqrt((-4)^(2)+(-6)^(2))`
`=sqrt(16+36)`
`=sqrt(52)`
`=sqrt(2xx2xx13)`
`=2sqrt(13)`........4
In `square PQRS`
`PQ=RS`.....[From 1 and 3]
`QR=PS` .....[From 2 and 4 ]
`:.square PQRS` is a prallelogram ...........(A quadrilateral is a parallelogram if its opposite sides are equal)
By distance formula,
`PR=sqrt((-5-2)^(2)+(-3-1)^(2))`
`=sqrt((-7)^(2)+(-4)^(2))`
`=sqrt(49+16)`
`=sqrt(65)`.............5
`QS=sqrt([-2-(-1)]^(2)+(-5-3)^(2))`
`=sqrt((-1)^(2)+(-8)^(2))`
`=sqrt(1+64)`
`=sqrt(65)`.....6
In parallelogram PQRS
`PR=QS` .......[From 5 and 6]
`:.square PQRS` is a rectangle ..........(A parallelogram is a rectangle, if its diagonals are equal)
`P(2,1),Q(-1,3),R(-5,-3)`and `S(-2,-5)` are the vertices of a reactangle.