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Trisection points and finding the equation of a line

Source: 1970 AHSME Problem 15

March 24, 2014
analytic geometryarithmetic sequenceAMC

Problem Statement

Lines in the xy-plane are drawn through the point (3,4)(3,4) and the trisection points of the line segment joining the points (−4,5)(-4,5) and (5,−1).(5,-1). One of these lines has the equation
<spanclass=′latex−bold′>(A)</span>3x−2y−1=0<spanclass=′latex−bold′>(B)</span>4x−5y+8=0<spanclass=′latex−bold′>(C)</span>5x+2y−23=0<span class='latex-bold'>(A) </span>3x-2y-1=0\qquad<span class='latex-bold'>(B) </span>4x-5y+8=0\qquad<span class='latex-bold'>(C) </span>5x+2y-23=0\qquad
<spanclass=′latex−bold′>(D)</span>x+7y−31=0<spanclass=′latex−bold′>(E)</span>x−4y+13=0<span class='latex-bold'>(D) </span>x+7y-31=0\qquad <span class='latex-bold'>(E) </span>x-4y+13=0
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