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Question

The points A(5,-1,1),B(7,-4,7),C(1,-6,10) and D(-1,-3,4) are vertices of a


A

Square

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B

Rhombus

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C

Rectangle

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D

None of these

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Solution

The correct option is B

Rhombus


Identify the quadrilateral formed by the given points

Given : A(5,-1,1),B(7,-4,7),C(1,-6,10) and D(-1,-3,4)

AB→=7-52+-4+12+(7-1)2⇒AB→=4+9+36⇒AB→=7

BC→=1-72+-6+42+(10-7)2⇒BC→=36+4+9⇒BC→=7

CD→=-1-12+-3+62+(4-10)2⇒CD→=4+9+36⇒CD→=7

DA→=5+12+-1+32+(1-4)2⇒DA→=36+4+9⇒DA→=7

Now,

AB→.BC→=(5×7)+(-1)×(-4)+(1×7)⇒AB→.BC→=35+4+7⇒AB→.BC→=46

Since AB→=BC→=CD→=DA→ but AB→.BC→≠0.

It shows that all sides are of equal length but the angle between the sides is not equal to 90°.

Therefore, given points cannot form the vertices of a square or rectangle but only of a rhombus.

Hence option B, Rhombus is the correct answer.


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