[Math] How to divide a line into 3 equal parts

analytic geometryeuclidean-geometrygeometryvectors

Find the points P1(x1,y1) and P2(x2,y2) on the line segment joining A(2,-1) and B(6,5) that divide the line segment into 3 equal parts.

I wasn’t able to figure out how to make this line into 3 parts. I got as far as finding the distance between the two being sqrt(52) and the midpoint being (4,2). Is there a certain way to split it into thirds instead of halves?

Best Answer

Guide:

  • Find vector $\vec{AB}$.
  • Then $\vec{OP_1} = \vec{OA} + \frac13\vec{AB}$
  • $\vec{OP_2} = \vec{OB} - \frac13\vec{AB}=\vec{OP_1} + \frac13\vec{AB}$
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