1.

ABCD is a rectangle formed by the points `A(1, 1),``B(1, 4),`` C(5, 4)` and `D(5, 1)`. P, Q, R and S are the midpoints of AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square? A rectangle? or a rhombus? Justify your answer.

Answer» Here,Coordinates of `P = ((-1-1)/2,(-1+4)/2) = (-1,3/2)`
Coordinates of `Q = ((-1+5)/2,(4+4)/2) = (2,4)`
Coordinates of `R = ((5+5)/2,(4-1)/2) = (5,3/2)`
Coordinates of `S = ((5-1)/2,(-1-1)/2) = (2,-1)`
So, `PQ = sqrt(3^2+(5/2)^2)=sqrt61/2`
`QR = sqrt(3^2+(-5/2)^2) = sqrt61/2`
`RS = sqrt((-3)^2+(-5/2)^2) = sqrt61/2`
`SP = sqrt((3)^2+(-5/2)^2) = sqrt61/2`
It means, all sides are `equal`.
Now, we check diagonals.
`PR = sqrt(6^2+0) = 6`
`QS = sqrt(0+(-5)^2) = 5`
It means, diagonals are not equal.
As all sides are equal and diagonals are not equal, `PQRS` is a rhombus.


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