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Practice Problems

\(\textbf{1)}\) Are the following points collinear? \((3,5),(6,7), \text{ and } (12,11)\)

Show Answer

\(\text{Yes, the points are collinear.}\)

Show Work

\(\text{Find the equation of the line through first two points.}\)

\(\,\,\,\,\,\,\text{Slope}\Rightarrow m=\displaystyle\frac{y_1-y_2}{x_1-x_2}\)
\(\,\,\,\,\,\,\,\,\,\,\,\,m=\displaystyle\frac{7-5}{6-3}\)
\(\,\,\,\,\,\,\,\,\,\,\,\,m=\displaystyle\frac{2}{3}\)

\(\,\,\,\,\,\,\text{Point Slope Formula} \Rightarrow y-y_1=m\left(x-x_1\right)\)
\(\,\,\,\,\,\,\,\,\,\,\,\, y-5=\frac{2}{3}\left(x-3\right)\)
\(\,\,\,\,\,\,\,\,\,\,\,\, y-5=\frac{2}{3}x-2\)
\(\,\,\,\,\,\,\,\,\,\,\,\, y=\frac{2}{3}x+3\)

\(\text{Check to see if 3rd point is on the same line.}\)

\(\,\,\,\,\,\,\,\,\,\,\,\, (11)=\frac{2}{3}(12)+3\)
\(\,\,\,\,\,\,\,\,\,\,\,\, 11=8+3\)
\(\,\,\,\,\,\,\,\,\,\,\,\, 11=11\)

\(\text{Yes, the points are collinear.}\)

\(\textbf{2)}\) Are the following points collinear? \((2,0),(5,3),\text{ and } (9,6)\)

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\(\text{No, the points are not collinear.}\)

Show Work

\(\text{Find the equation of the line through first two points.}\)

\(\,\,\,\,\,\,\text{Slope}\Rightarrow m=\displaystyle\frac{y_1-y_2}{x_1-x_2}\)
\(\,\,\,\,\,\,\,\,\,\,\,\,m=\displaystyle\frac{3-0}{5-2}\)
\(\,\,\,\,\,\,\,\,\,\,\,\,m=\displaystyle\frac{3}{3}=1\)

\(\,\,\,\,\,\,\text{Point Slope Formula} \Rightarrow y-y_1=m\left(x-x_1\right)\)
\(\,\,\,\,\,\,\,\,\,\,\,\, y-0=1\left(x-2\right)\)
\(\,\,\,\,\,\,\,\,\,\,\,\, y=x-2\)

\(\text{Check to see if 3rd point is on the same line.}\)

\(\,\,\,\,\,\,\,\,\,\,\,\, (6)=1(9)-2\)
\(\,\,\,\,\,\,\,\,\,\,\,\, 6\ne 7\)

\(\text{No, the points are not collinear.}\)

\(\textbf{3)}\) Are the following points collinear? \((-2,6),(2,-2), \text{ and } (5,-8)\)

Show Answer

\(\text{Yes, the points are collinear.}\)

Show Work

\(\text{Find the equation of the line through first two points.}\)

\(\,\,\,\,\,\,\text{Slope}\Rightarrow m=\displaystyle\frac{y_1-y_2}{x_1-x_2}\)
\(\,\,\,\,\,\,\,\,\,\,\,\,m=\displaystyle\frac{(-2)-6}{2-(-2)}\)
\(\,\,\,\,\,\,\,\,\,\,\,\,m=\displaystyle\frac{-8}{4}=-2\)

\(\,\,\,\,\,\,\text{Point Slope Formula} \Rightarrow y-y_1=m\left(x-x_1\right)\)
\(\,\,\,\,\,\,\,\,\,\,\,\, y-(-2)=-2\left(x-2\right)\)
\(\,\,\,\,\,\,\,\,\,\,\,\, y+2=-2x+4\)
\(\,\,\,\,\,\,\,\,\,\,\,\, y=-2x+2\)

\(\text{Check to see if 3rd point is on the same line.}\)

\(\,\,\,\,\,\,\,\,\,\,\,\, (-8)=-2(5)+2\)
\(\,\,\,\,\,\,\,\,\,\,\,\, -8=-10+2\)
\(\,\,\,\,\,\,\,\,\,\,\,\, -8=-8\)

\(\text{Yes, the points are collinear.}\)

\(\textbf{4)}\)Are the following points collinear? \((-3,5),(0,0),\text{ and } (5,-3)\)

Show Answer

\(\text{No, the points are not collinear.}\)

Show Work

\(\text{Find the equation of the line through first two points.}\)

\(\,\,\,\,\,\,\text{Slope}\Rightarrow m=\displaystyle\frac{y_1-y_2}{x_1-x_2}\)
\(\,\,\,\,\,\,\,\,\,\,\,\,m=\displaystyle\frac{0-5}{0-(-3)}\)
\(\,\,\,\,\,\,\,\,\,\,\,\,m=\displaystyle\frac{-5}{3}\)

\(\,\,\,\,\,\,\text{Point Slope Formula} \Rightarrow y-y_1=m\left(x-x_1\right)\)
\(\,\,\,\,\,\,\,\,\,\,\,\, y-0=-\frac{5}{3}\left(x-0\right)\)
\(\,\,\,\,\,\,\,\,\,\,\,\, y=-\frac{5}{3}x\)

\(\text{Check to see if 3rd point is on the same line.}\)

\(\,\,\,\,\,\,\,\,\,\,\,\, (-3)=-\frac{5}{3}(5)\)
\(\,\,\,\,\,\,\,\,\,\,\,\, -3\ne -\frac{25}{3}\)

\(\text{No, the points are not collinear.}\)

\(\textbf{5)}\) Are the following points collinear? \((-2,7),(0,0), \text{ and } (2,-7)\)

Show Answer

\(\text{Yes, the points are collinear.}\)

Show Work

\(\text{Find the equation of the line through first two points.}\)

\(\,\,\,\,\,\,\text{Slope}\Rightarrow m=\displaystyle\frac{y_1-y_2}{x_1-x_2}\)
\(\,\,\,\,\,\,\,\,\,\,\,\,m=\displaystyle\frac{0-7}{0-(-2)}\)
\(\,\,\,\,\,\,\,\,\,\,\,\,m=\displaystyle\frac{-7}{2}\)

\(\,\,\,\,\,\,\text{Point Slope Formula} \Rightarrow y-y_1=m\left(x-x_1\right)\)
\(\,\,\,\,\,\,\,\,\,\,\,\, y-0=-\frac{7}{2}\left(x-0\right)\)
\(\,\,\,\,\,\,\,\,\,\,\,\, y=-\frac{7}{2}x\)

\(\text{Check to see if 3rd point is on the same line.}\)

\(\,\,\,\,\,\,\,\,\,\,\,\, (-7)=-\frac{7}{2}(2)\)
\(\,\,\,\,\,\,\,\,\,\,\,\, -7=-7\)

\(\text{Yes, the points are collinear.}\)

\(\textbf{6)}\) True or False, “Any 2 points are collinear.”

Show Answer

\(\text{True}\)

\(\textbf{7)}\) True or False, “Any 3 points are collinear.”

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\(\text{False}\)

\(\textbf{8)}\) True or False, “3 points can be collinear.”

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\(\text{True}\)

See Related Pages\(\)

\(\bullet\text{ Geometry Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)

\(\bullet\text{ Graphing Linear Equations}\)
\(\,\,\,\,\,\,\,\,2x-3y=6 \)

\(\bullet\text{ Slope Formula}\)
\(\,\,\,\,\,\,\,\,m=\frac{y_2-y_1}{x_2-x_1}\)

\(\bullet\text{ Net Change}\)
\(\,\,\,\,\,\,\,\,y_2-y_1\)

\(\bullet\text{ Slope Intercept Form}\)
\(\,\,\,\,\,\,\,\,y=mx+b\)

\(\bullet\text{ Parallel and Perpendicular Slope}\)
\(\,\,\,\,\,\,\,\,m_1=m+2,\,\,\,m_1=\frac{1}{m_2}\)

\(\bullet\text{ Distance Between a Point and a Line}\)
\(\,\,\,\,\,\,\,\,(3,4) \text{ and } y=\frac{3}{4}x−2\)

\(\bullet\text{Finding x- and y- intercepts}\)
\(\,\,\,\,\,\,\,\,y=2x+4\)

In Summary

Collinear points are points that lie on the same line. Any 2 points are collinear. 3 or more points may or may not be collinear. If you can draw a straight line through the points, they are collinear.

A good method to check if multiple points are collinear on the coordinate plane is to see if all the points are solutions to the same linear equation. If they are, then the points are collinear.

Collinear points are usually introduced in a high school geometry class, but could be introduced earlier, and they have applications in higher levels of math.

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