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 ### 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

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

${m}_{AB} ={m}_{BC} ={m}_{AC}$

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 ### 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

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

${d}_{AB} + {d}_{BC} = {d}_{AC}$
therefore points $$A$$, $$B$$ and $$C$$ lie on the same straight line.