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 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 midpoint of side AB

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

Similarly, the coordinates of Q, R and S are `(2, 4),(5, 3/2)` and (2, –1) respectively

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.