`A B C D` is a rectangle formed by joining the points `A(-1,\ -1),\ \ B(-1,\ 4),\ \ C(5,\ 4)` and `D(5,\ -1)dot` `P ,\ Q ,\ R` and `S` are the mid-points of sides `A B ,\ B C ,\ C D` and `D A` respectively. Is the quadrilateral `P Q R S` a square? a rectangle? or a rhombus? Justify your answer.

Explanation

Given vertices of a rectangle are A(-1, -1), B(-1, 4), C(5, 4) and D(5, -1).
`therefore` Mid-point of AB is `P((-1-1)/(2), (-1+4)/(2))or P(-1, (3)/(2)).`
`" "[because "mid-point"=((x_(1)+x_(2))/(2), (y_(1)+y_(2))/(2))]`
Mid-point of BC is `" "Q((5-1)/(2), (4+4)/(2)) or Q(2,4)`
Mid-point of CD is `" "R((5+5)/(2), (4-1)/(2))or R(5, (3)/(2))`

Mid-point of AD is `" "S((-1+5)/(2), (-1-1)/2)or S(2, 1)`.
Now , `" "PQ=sqrt((2+1)^(2)+(4-(3)/(2))^(2))=sqrt(3^(2)+((5)/(2))^(2))=sqrt(9+(25)/(4))=sqrt((36+25)/(4))=sqrt((61)/(4))`
`" "QR=sqrt((5-1)^(2)+((3)/(2)-4)^(2))=sqrt(3^(2)+((5)/(2))^(2))=sqrt(9+(25)/(4))=sqrt((36+25)/(4))=sqrt((61)/(4))`
`" "RS=sqrt((2-5)^(2)+(-1-(3)/(2))^(2))=sqrt((-3)^(2)+(-(5)/(2))^(2))=sqrt(9+(25)/(4))=sqrt((36+25)/(4))=sqrt((61)/(4))`
`" "SP=sqrt((-1-2)^(2)+((3)/(2)+1)^(2))=sqrt((-3)^(2)+((5)/(2))^(2))=sqrt(9+(25)/(4))=sqrt((36+25)/(4))=sqrt((61)/(4))`
`because PQ=QR=RS=SP`
Now, `PR=sqrt((5+1)^(2)+((3)/(2)-(3)/(2))^(2))=sqrt(6^(2)+0)=6`
and `SQ=sqrt((2-2)^(2)+(4+1)^(2))=sqrt(0+5^(2))=5`
`rArr" "PR ne SQ`
Since, all the sides are equal but diagonals are not equal.
`thereforePQRS` is rhombus.