Question

# If P(2,−1),Q(3,4),R(−2,3) and S(−3,−2) be four points in a plane, show that PQRS is a rhombus but not a square. Find the area of the rhombus.

Solution
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#### The given points are P(2,-1), Q(3,4), R(-2,3) and S(-3,-2).We havePQ=√(3−2)2+(4+1)2=√12+52=√26unitsQR=√(−2−3)2+(3−4)2=√25+1=√26unitsRS=√(−3+2)2+(−2−3)2=√1+25=√26unitsSP=√(−3−2)2+(−2−3)2=√26unitsPR=√(−2−2)2+(3+1)2=√16+16=4√2unitsand, QS=√(−3−3)2+(−2−4)2=√36+36=6√2units∴PQ=QR=RS=SP=√26unitsand, PR≠QSThis means that PQRS is quadrilateral whose sides are equal but diagonals are not equal. Thus, PQRS is a rhombus but not a square..Now, Area of rhombus PQRS=12×(Productoflengthsofdiagonals)⇒AreaofrhombusPQRS=12×(PR×QS)⇒AreaofrhombusPQRS=(12×4√2×6√2)sq.units=24sq.units

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