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 = \(\frac{x_1+x_2}2\), y = \(\frac{y_1+y_2}2\)
For midpoint P of side AB,
x = \(\frac{-1-1}2\), y = \(\frac{-1+4}2\)
x = -1 , y = \(\frac{3}2\)
Hence, the coordinates of P are (-1 ,\(\frac{3}2\) )
For midpoint Q of side BC,
x = \(\frac{-1+5}2\), y = \(\frac{4+4}2\)
x = 2 , y = 4
Hence,
the coordinates of Q are (2 ,4)
For midpoint R of side CD,
x = \(\frac{5+5}2\), y = \(\frac{-1+4}2\)
x = 5 , y = \(\frac{3}2\)
Hence, the coordinates of R are (5 ,\(\frac{3}2\) )
For midpoint S of side AD,
x = \(\frac{-1+5}2\), y = \(\frac{-1-1}2\)
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,
For RS,
Here we can observe that all lengths of □PQRS are equal.
Now for diagonal PR,
= 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 .