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# 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 mid-points of AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square? a rectangle? or a rhombus? Justify your answer.

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Solution

## Let P,Q,R and S are the midpoint of AB,BC,CD andDA.∴ Co-ordinates of P=[x1+x22,y1+y22]=[−1−12,−1+42]=[−1,32]∴ Co-ordinates of Q=[x1+x22,y1+y22]=[−1+52,4+42]=[2,4]∴ Co-ordinates of R=[x1+x22,y1+y22]=[5+52,4−12]=[5,32]∴ Co-ordinates of S=[x1+x22,y1+y22]=[5−12,−1−12]=[4,−1]Now, length of PQ=√(−1−2)2+(32−4)2=√(−3)2+(−52)2=√9+254=√614Length of QR=√(2−5)2+(4−32)2=√(−3)2+(52)2=√9+254=√614Length of RS=√(5−2)2+(32+1)2=√(3)2+(52)2=√9+254=√614Length of S=√(2+1)2+(−1−32)2=√(3)2+(−52)2=√9+254=√614Length of diagonal PR=√(−1−5)2+(32−32)2=√(62)=√36=6Length of diagonal QS=√(2−2)2+(4+1)2=√(5)2=√25=5Hence the all the sides of the quadrilateral PQRS are equal but the diagonals are not equal then PQRS is a rhombus.   Suggest Corrections  0      Similar questions  Related Videos   Slope of a Line
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