Sponsor Area

Sequences And Series

Question
CBSEENMA11014971

Show that A (a, b, c), B (b, c, a) and C (c, a, b) are vertices of an equilateral triangle.

Solution
straight A space left parenthesis straight a comma space straight b space comma straight c right parenthesis left right arrow 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 straight b comma space straight c comma space straight a right parenthesis left right arrow 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 straight c comma space straight a comma space straight b right parenthesis left right arrow space left parenthesis straight x subscript 3 comma space straight y subscript 3 comma space straight z subscript 3 right parenthesis
Using distance formula, we have
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#6 {main}</pre>
                             equals square root of left parenthesis straight a minus straight b right parenthesis squared plus left parenthesis straight b minus straight c right parenthesis squared plus left parenthesis straight c minus straight a right parenthesis squared end root
                              equals space square root of 2 left parenthesis straight a squared plus straight b squared plus straight c squared minus ab minus bc minus ca right parenthesis end root              ...(i)
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#6 {main}</pre>
                              equals space square root of left parenthesis straight b minus straight c right parenthesis squared plus left parenthesis straight c minus straight a right parenthesis squared plus left parenthesis straight a minus straight b right parenthesis squared end root
                               equals space square root of 2 left parenthesis straight a squared plus straight b squared plus straight c squared minus ab minus bc minus ca right parenthesis end root              ...(ii)
                     space space space CA equals square root of left parenthesis straight x subscript 3 minus straight x subscript 1 right parenthesis squared plus left parenthesis straight y subscript 3 minus straight y subscript 1 right parenthesis squared plus left parenthesis straight z subscript 3 minus straight z subscript 1 right parenthesis squared end root
                            equals space square root of left parenthesis straight c minus straight a right parenthesis squared plus left parenthesis straight a minus straight b right parenthesis squared plus left parenthesis straight b minus straight c right parenthesis squared end root
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#6 {main}</pre>                       ...(iii)
From (i), (ii) and (iii), we have
                            AB = BC = CA
Hence, triangle ABC is an equilateral triangle.