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Points P

Source: AHSME 1963 Problem 37

January 13, 2014
AMC

Problem Statement

Given points P1,P2,⋯ ,P7P_1, P_2,\cdots,P_7 on a straight line, in the order stated (not necessarily evenly spaced). Let PP be an arbitrarily selected point on the line and let ss be the sum of the undirected lengths PP1,PP2,⋯ ,PP7.PP_1, PP_2, \cdots , PP_7. Then ss is smallest if and only if the point PP is:
<spanclass=′latex−bold′>(A)</span> midway between P1 and P7<spanclass=′latex−bold′>(B)</span> midway between P2 and P6<spanclass=′latex−bold′>(C)</span> midway between P3 and P5<span class='latex-bold'>(A)</span>\ \text{midway between }P_1\text{ and }P_7\qquad <span class='latex-bold'>(B)</span>\ \text{midway between }P_2\text{ and }P_6\qquad <span class='latex-bold'>(C)</span>\ \text{midway between }P_3\text{ and }P_5 \qquad
<spanclass=′latex−bold′>(D)</span> at P4<spanclass=′latex−bold′>(E)</span> at P1 <span class='latex-bold'>(D)</span>\ \text{at }P_4 \qquad <span class='latex-bold'>(E)</span>\ \text{at }P_1
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