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Sequences And Series

Question
CBSEENMA11014974

Verify the following:
(-1, 2, 1), (1, -2, 5), (4, -7, 8) and (2, -3, 4) are the vertices of a parallelogram.

Solution
straight A space left parenthesis negative 1 comma space 2 comma space 1 right parenthesis space left right arrow space left parenthesis straight x subscript 1 comma space straight y subscript 1 comma space straight z subscript 1 right parenthesis
straight B space left parenthesis 1 comma space minus 2 comma space 5 right parenthesis space left right arrow space left parenthesis straight x subscript 2 comma space straight y subscript 2 comma space straight z subscript 2 right parenthesis
straight C space left parenthesis 4 comma space minus 7 comma space 8 right parenthesis space left right arrow space left parenthesis straight x subscript 3 comma space straight y subscript 3 comma space straight z subscript 3 right parenthesis
straight D space left parenthesis 2 comma space minus 3 comma space 4 right parenthesis space left right arrow space left parenthesis straight x subscript 4 comma space straight y subscript 4 comma space straight z subscript 4 right parenthesis
Using distance formula, we have
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#6 {main}</pre>  ...(i)
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#6 {main}</pre>
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                             space space space space equals space square root of left parenthesis 4 minus 2 right parenthesis squared plus left parenthesis negative 7 plus 3 right parenthesis squared plus left parenthesis 8 minus 4 right parenthesis squared end root space equals space square root of 4 plus 16 plus 16 end root space equals space square root of 36 space equals space 6   ...(iii)
                          DA space equals space square root of left parenthesis straight x subscript 4 minus straight x subscript 1 right parenthesis squared plus left parenthesis straight y subscript 4 minus straight y subscript 1 right parenthesis squared plus left parenthesis straight z subscript 4 minus straight z subscript 1 right parenthesis squared end root
                              space space space equals space square root of left parenthesis 2 plus 1 right parenthesis squared plus left parenthesis negative 3 minus 2 right parenthesis squared plus left parenthesis 4 minus 1 right parenthesis squared end root space equals space square root of 9 plus 25 plus 9 end root space equals space square root of 43             ...(iv)
From (i), (ii), (iii) and (iv), we have AB = CD, BC = DA
Hence, ABCD is a parallelogram.