Question

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.

Answer 1

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 .