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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.
Solution
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.
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