Question

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 1

P is the mid point of side AB

THerefore the coordinates of P are ((-1-1)/2,(-1+4)/2) = (-1, 3/2)

Similary the coordinates of Q , R and S are  (2,4),(5, 3/2), and (2, -1) respectilvely

Length of PQ =`sqrt((-1-2)^2 + (3/2-4)^2) =sqrt(9+25/4)= sqrt(61/4)`

Length of QR =`sqrt((2-5)^2 + (4-3/2)^2) =sqrt(9+25/4)= sqrt(61/4)`

Length of RS =`sqrt((5-2)^2 + (3/2+1)^2) =sqrt(9+25/4)= sqrt(61/4)`

Length of SP =`sqrt((2+1)^2 + (-1-3/2)^2) =sqrt(9+25/4)= sqrt(61/4)`

Length of PR =`sqrt((-1-5)^2 + (3/2-3/2)^2) = 6`

Length of QS = `sqrt((2-2)^2 + (4+1)^2) = 5`

It can be observed that all sides of the given quadrilateral are of the same measure. However, the diagonals are of different lengths. Therefore, PQRS is a rhombus.