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Printjmc
algebra intermediate
Problem
Of the four points , , , and , three lie on the same line. Which point is not on the line?
Solution
Consider points , , and . If the slope between point and point is the same as the slope between point and point , , , and are collinear. So we must find the slopes between every pair of possible points. Let us name the points: , , , and . We make a chart of all possible pairs of points and calculate the slope:
\begin{array}{c|c} Points& Slope \\ \hline \vspace{0.05in} A,B&$\frac{11-2}{9-2}=\frac{9}{7}$\\ \vspace{0.05in} $A,C%%DISP_0%%amp;$\frac{7-2}{5-2}=\frac{5}{3}$\\ \vspace{0.05in} $A,D%%DISP_0%%amp;$\frac{17-2}{11-2}=\frac{15}{9}=\frac{5}{3}$\\ \vspace{0.05in} $B,C%%DISP_0%%amp;$\frac{7-11}{5-9}=\frac{-4}{-4}=1$\\ \vspace{0.05in} $B,D%%DISP_0%%amp;$\frac{17-11}{11-9}=\frac{6}{2}=3$\\ \vspace{0.05in} $C,D%%DISP_0%%amp;$\frac{17-7}{11-5}=\frac{10}{6}=\frac{5}{3}$ \end{array}As we can see, the slopes between and , and , and and are the same, so , , and lie on a line, Thus , or the point , is not on the line.
\begin{array}{c|c} Points& Slope \\ \hline \vspace{0.05in} A,B&$\frac{11-2}{9-2}=\frac{9}{7}$\\ \vspace{0.05in} $A,C%%DISP_0%%amp;$\frac{7-2}{5-2}=\frac{5}{3}$\\ \vspace{0.05in} $A,D%%DISP_0%%amp;$\frac{17-2}{11-2}=\frac{15}{9}=\frac{5}{3}$\\ \vspace{0.05in} $B,C%%DISP_0%%amp;$\frac{7-11}{5-9}=\frac{-4}{-4}=1$\\ \vspace{0.05in} $B,D%%DISP_0%%amp;$\frac{17-11}{11-9}=\frac{6}{2}=3$\\ \vspace{0.05in} $C,D%%DISP_0%%amp;$\frac{17-7}{11-5}=\frac{10}{6}=\frac{5}{3}$ \end{array}As we can see, the slopes between and , and , and and are the same, so , , and lie on a line, Thus , or the point , is not on the line.
Final answer
(9,11)