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# The points $(5,0,2),\:(2,-6,0),\:(4,-9,6)\:and\:(7,-3,8)$ represent vertices of a ?

$(a)\:Square\:\:\qquad\:(b)\:Rhombus\:\:\qquad\:(c)\:Rectangle\:\:\qquad\:(d)\:Parallelogram.$

Can you answer this question?

Let the points be $A(5,0,2),\:B(2,-6,0),\:C(4,-9,6),\:D(7,-3,8)$
$AB=\sqrt {9+36+4}=7$
$BC=\sqrt{4+9+36}=7$
$BD=\sqrt {9+36+4}=7$
$AD=\sqrt{ 4+9+36}=7$
$\overrightarrow {AB}=(-3,-6,-2)$
$\overrightarrow {AD}=(2,-3,6)$
$\overrightarrow {AB}.\overrightarrow {AD}=(-3,-6,-2).(2,-3,6)=-6+18-12=0$
Since all the four sides are equal and angle between adjacent sides is $90^{\circ}$,
$ABCD$ is a square.
answered Jan 11, 2014