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

Here given that A (-1, -1), B (-1, 4), C (5, 4) and D (5,-1).Also P, Q, R and S are the mid-points of sides AB, BC, CD and DA respectively.

By midpoint formula.

x = , y =

For midpoint P of side AB,

x = , y =

x = -1 , y =

Hence, the coordinates of P are (-1 , )

For midpoint Q of side BC,

x = , y =

x = 2 , y = 4

Hence, the coordinates of Q are (2 ,4)

For midpoint R of side CD,

x = , y =

x = 5 , y =

Hence, the coordinates of R are (5 , )

For midpoint S of side AD,

x = , y =

x = 2 , y = -1

Hence, the coordinates of S are (2 ,-1)

Now we find length of the length of the □PQRS,

By distance formula,

XY =

For PQ,

PQ =

=

= units

For QR,

QR =

=

= units

For RS,

RS =

=

= units

For PS,

PS =

=

= units

Here we can observe that all lengths of □PQRS are equal.

Now for diagonal PR,

PR =

=

= units

Now for diagonal QS,

QS =

=

= 5 units

Here in □PQRS, diagonals are unequal.

We know that a quadrilateral whose all sides are equal and diagonals are unequal, it is a rhombus.

Hence, our □PQRS is rhombus .