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Question
Verify the following:
(5, –1, 1), (7, –4,7), (1, –6,10) and (–1, – 3,4) are the vertices of a rhombus.
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Solution
Let A(5,\[-\]1, 1) , B(7, \[-\]4, 7) , C(1, \[-\]6, 10), D(\[-\]1, \[-\]3, 4) be the vertices of quadrilateral \[\square ABCD\]
\[AB = \sqrt{\left( 5 - 7 \right)^2 + \left( - 1 + 4 \right)^2 + \left( 1 - 7 \right)^2}\]
\[ = \sqrt{4 + 9 + 36}\]
\[ = \sqrt{49}\]
\[ = 7\]
\[BC = \sqrt{\left( 7 - 1 \right)^2 + \left( - 4 + 6 \right)^2 + \left( 7 - 10 \right)^2}\]
\[ = \sqrt{36 + 4 + 9}\]
\[ = \sqrt{49}\]
\[ = 7\]
\[CD = \sqrt{\left( 1 + 1 \right)^2 + \left( - 6 + 3 \right)^2 + \left( 10 - 4 \right)^2}\]
\[ = \sqrt{4 + 9 + 36}\]
\[ = \sqrt{49}\]
\[ = 7\]
\[DA = \sqrt{\left( - 1 - 5 \right)^2 + \left( - 3 + 1 \right)^2 + \left( 4 - 1 \right)^2}\]
\[ = \sqrt{36 + 4 + 9}\]
\[ = \sqrt{49}\]
\[ = 7\]
\[ \therefore AB = BC = CD = DA\]
Since, all the sides are equal.
Thus, quadrilateral ABCS is a rhombus.
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