Question

# Question 8 ABCD is a rectangle formed by points A (-1,-1), B(-1,4), C(5,4) and D(5,-1). P, Q, R, and S are mid-points of AB, BC, CD, and DA respectively. Is the quadrilateral PQRS a square, rectangle or rhombus? Justify your answer.

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Solution

## Coordinates of P can be calculated as follows: [Since, mid-point of the line segment joining the points (x1,y1) and (x2,y2) is (x1+x22,y1+y22)] (−1−12,4−12)=(−1,32) Coordinates of Q can be calculated as follows: (5−12,4+42)=(2,4) Coordinates of R can be calculated as follows: (5+52,4−12)=(5,32) Coordinates of S can be calculated as follows: (5−12,−1−12)=(2,−1) Length of PQ can be calculated as follows: PQ=√(2+1)2+(4−32)2 =√(2+1)2+(52)2 =√(9)+(254) =√612 Similarly, QR can be calculated as follows: QR =√(5−2)2+(32−4)2 =√32+(−52)2 =√612 Similarly, PS =√(2+1)2+(−1−32)2 =√32+(−52)2 =√612 Similarly, QR can be calculated as follows: SR =√(2−5)2+(−1−32)2 =√(−3)2+(−52)2 =√612 The above values shows that PQ = QR = RS = PS, i.e. all sides are equal. Now let us calculate the diagonals. PR=√(5+1)2+(32−32)2 =√62=6 QS=√(2−2)2+(−1−4)2 =√(−5)2=5 Hence, it is clear that while all sides are equal, diagonals are not equal. So, the given figure is a rhombus.

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