Mathematics » Analytical Geometry » Gradient Of A Line

Points On a Line

Points On a Line

A straight line is a set of points with a constant gradient between any of the two points. There are two methods to prove that points lie on the same line: the gradient method and a method using the distance formula.

Fact:

If two points lie on the same line then the two points are said to be collinear.

Example

Question

Prove that \(A(-3;3)\), \(B(0;5)\) and \(C(3;7)\) are on a straight line.

Draw a sketch

034b7141a42427555bbe1f8f110f53b6.png

Calculate two gradients between any of the three points

\begin{align*} m & = \cfrac{{y}_{2} – {y}_{1}}{{x}_{2} – {x}_{1}} \\ {m}_{AB} & = \cfrac{5 – 3}{0 – (-3)} = \cfrac{2}{3} \end{align*}

and

\[{m}_{BC} = \cfrac{7 – 5}{3 – 0} = \cfrac{2}{3}\]

OR

\[{m}_{AC} = \cfrac{3-7}{3-3} = \cfrac{-4}{-6} = \cfrac{2}{3}\]

and

\[{m}_{BC} = \cfrac{7-5}{3-0} = \cfrac{2}{3}\]

Explain your answer

\[{m}_{AB} ={m}_{BC} ={m}_{AC}\]

Therefore the points \(A\), \(B\) and \(C\) are on a straight line.

To prove that three points are on a straight line using the distance formula, we must calculate the distances between each pair of points and then prove that the sum of the two smaller distances equals the longest distance.

Example

Question

Prove that \(A(-3;3)\), \(B(0;5)\) and \(C(3;7)\) are on a straight line.

Draw a sketch

5ae61be3fb148f843425403e848c29a6.png

Calculate the three distances \(AB\), \(BC\) and \(AC\)

\begin{align*} {d}_{AB} & = \sqrt{{(-3 – 0)}^{2} + {(3-5)}^{2}} \\ & = \sqrt{{(-3)}^{2} + {(-2)}^{2}} \\ & = \sqrt{9 + 4} \\ & = \sqrt{13} \\ \\ {d}_{BC} & = \sqrt{{(0 – 3)}^{2} + {(5 – 7)}^{2}} \\ & = \sqrt{{(-3)}^{2} + {(-2)}^{2}} \\ & = \sqrt{9 + 4} \\ & = \sqrt{13} \\ \\ {d}_{AC} & = \sqrt{{((-3) – 3)}^{2} + {(3 – 7)}^{2}} \\ & = \sqrt{{(-6)}^{2} + {(-4)}^{2}} \\ & = \sqrt{36+16} \\ & = \sqrt{52} \end{align*}

Find the sum of the two shorter distances

\[{d}_{AB} + {d}_{BC} = \sqrt{13} + \sqrt{13} = 2\sqrt{13} = \sqrt{4\times 13} = \sqrt{52}\]

Explain your answer

\[{d}_{AB} + {d}_{BC} = {d}_{AC}\]

therefore points \(A\), \(B\) and \(C\) lie on the same straight line.

[Attributions and Licenses]


This is a lesson from the tutorial, Analytical Geometry and you are encouraged to log in or register, so that you can track your progress.

Log In

Share Thoughts