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Points of Trisection

Show that A 3...

Question

Show that A (3,-2) is a point of trisection of the line-segment joining the points (2, 1) and (5, -8). Also, find the co-ordinates of the other point of trisection.

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Solution

Let A and B be the point of trisection of the line segment joining the points P (2, 1) and Q (5, -8).
So, PA = AB = BQ

$W e space h a v e space P A colon A Q equals 1 colon 2 C o minus o r d i n a t e s space o f space t h e space p o i n t space A space a r e open parentheses fraction numerator 1 cross times 5 plus 2 cross times 2 over denominator 1 plus 2 end fraction comma fraction numerator 1 cross times left parenthesis negative 8 right parenthesis plus 2 cross times 1 over denominator 1 plus 2 end fraction close parentheses equals open parentheses 9 over 3 comma fraction numerator negative 6 over denominator 3 end fraction close parentheses equals open parentheses 3 comma negative 2 close parentheses h e n c e space comma A left parenthesis 3 comma negative 2 right parenthesis space i s space a space p o i n t space o f space t r i s e c t i o n space o f space P Q W e space h a v e space P B colon B Q equals 2 colon 1 C o minus o r d i n a t e s space o f space t h e space p o i n t space b space a r e open parentheses fraction numerator 2 cross times 5 plus 1 cross times 2 over denominator 2 plus 1 end fraction comma fraction numerator 2 cross times left parenthesis negative 8 right parenthesis plus 1 cross times 1 over denominator 2 plus 1 end fraction close parentheses equals open parentheses fraction numerator 10 plus 2 over denominator 3 end fraction comma fraction numerator negative 16 plus 1 over denominator 3 end fraction close parentheses equals open parentheses 4 comma negative 5 close parentheses$

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