A) form a rectangle which is not a square
B) form a rhombus which is not a square
C) form a parallelogram which is not a rhombus
D) are collinear
Correct Answer: B
Solution :
Given points are \[A(1,3,4),B(-1,6,10),C(-7,4,7)\]and \[D(-5,1,1).\] \[AB=\sqrt{{{(-1-1)}^{2}}+{{(6-3)}^{2}}+{{(10-4)}^{2}}}\] \[=\sqrt{{{(-2)}^{2}}+{{(3)}^{2}}+{{(6)}^{2}}}\] \[=\sqrt{4+9+36}=\sqrt{49}=7\] \[BC=\sqrt{{{(-7+1)}^{2}}+{{(4-6)}^{2}}+{{(7-10)}^{2}}}\] \[=\sqrt{{{(-6)}^{2}}+{{(-2)}^{2}}+{{(-3)}^{2}}}\] \[=\sqrt{36+4+9}=\sqrt{49}=7\] \[CD=\sqrt{{{(-5+7)}^{2}}+{{(1-4)}^{2}}+{{(1-7)}^{2}}}\] \[=\sqrt{{{(2)}^{2}}+{{(-3)}^{2}}+{{(-6)}^{2}}}\] \[=\sqrt{4+9+36}=7\] \[AD=\sqrt{{{(-5-1)}^{2}}+{{(1-3)}^{2}}+{{(1-4)}^{2}}}\] \[=\sqrt{{{(-6)}^{2}}+{{(-2)}^{2}}+{{(-3)}^{2}}}\] \[=\sqrt{36+4+9}=7\] \[AC=\sqrt{{{(-7-1)}^{2}}+{{(4-3)}^{2}}+{{(7-4)}^{2}}}\] \[=\sqrt{{{(-8)}^{2}}+{{(1)}^{2}}+{{(3)}^{2}}}\] \[=\sqrt{64+1+9}=\sqrt{74}\] \[BD=\sqrt{{{(-5+1)}^{2}}+{{(1-6)}^{2}}+{{(1-10)}^{2}}}\] \[=\sqrt{{{(-4)}^{2}}+{{(-5)}^{2}}+{{(-9)}^{2}}}\] \[=\sqrt{16+25+81}=\sqrt{112}\] Since, sides \[AB=BC=CD=AD=7\] and diagonals, \[AC\ne BD\] So, given points form a rhombus which is not a square.
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