Sequences And Series

Question
CBSEENMA11015017

Find the lengths of the medians of the triangle with vertices A (0, 0, 6), B (0, 4, 0) and C (6, 0, 0).

Solution
straight A space left parenthesis 0 comma space 0 comma space 6 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 0 comma space 4 comma space 0 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 6 comma space 0 comma space 0 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
D is mid-point of BC
rightwards double arrow D is open parentheses fraction numerator straight x subscript 2 plus straight x subscript 3 over denominator 2 end fraction comma space fraction numerator straight y subscript 2 plus straight y subscript 3 over denominator 2 end fraction comma space fraction numerator straight z subscript 2 plus straight z subscript 3 over denominator 2 end fraction close parentheses space left right arrow space open parentheses fraction numerator 0 plus 6 over denominator 2 end fraction comma space fraction numerator 4 plus 0 over denominator 2 end fraction comma space fraction numerator 0 plus 0 over denominator 2 end fraction close parentheses
                           = (3, 2, 0)
                                                                     
Using distance formula, we have
              AD equals square root of left parenthesis 0 minus 3 right parenthesis squared plus left parenthesis 0 minus 2 right parenthesis squared plus left parenthesis 6 minus 0 right parenthesis squared end root space equals space square root of 9 plus 4 plus 36 end root space equals space square root of 49 space equals space 7
E is the mid-point of CA.
rightwards double arrow                           straight E space equals space open parentheses fraction numerator straight x subscript 3 plus straight x subscript 1 over denominator 2 end fraction comma space fraction numerator straight y subscript 3 plus straight y subscript 1 over denominator 2 end fraction comma space fraction numerator straight z subscript 3 plus straight z subscript 1 over denominator 2 end fraction close parentheses space left right arrow space open parentheses fraction numerator 6 plus 0 over denominator 2 end fraction comma space fraction numerator 0 plus 0 over denominator 2 end fraction comma space fraction numerator 0 plus 6 over denominator 2 end fraction close parentheses space equals space left parenthesis 3 comma space 0 comma space 3 right parenthesis
∴                       BE space equals space square root of left parenthesis 0 minus 3 right parenthesis squared plus left parenthesis 4 minus 0 right parenthesis squared plus left parenthesis 0 minus 3 right parenthesis squared end root space equals space square root of 9 plus 16 plus 9 end root space equals space square root of 34
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