# Coordinates and ratio

## In a nutshell

Knowing the coordinates of the end-points of a line segment is enough to describe locations on the line segment. The midpoint is just halfway between the $x$- and $y$-coordinates, but other positions can be found using ratios.

## Midpoints

To find the point in the middle of two other points, find the middle between the points' $x$-coordinates and the middle between the points' $y$-coordinates. These middle values are the $x$- and $y$- coordinates of the midpoint.

Finding the middle between two numbers can be done by halving the difference between them and then adding this to the smaller of the two numbers. Or more formally, using the following procedure.

#### procedure

**1.** | Designate one of your points as point $A$ and the other as point $B$. Let point $A$ have the coordinates $(x_A,y_A)$ and point $B$ have the coordinates $(x_B,y_B)$. |

**2.** | Find the middle $x$-value using the formula: $x_A+\frac{x_B-x_A}{2}$ |

**3.** | Find the middle $y$-value using the similar formula: $y_A+\frac{y_B-y_A}{2}$ |

**4.** | You now have that the midpoint has coordinates $(x_A+\frac{x_B-x_A}{2},y_A+\frac{y_B-y_A}{2})$. |

## Using a ratio to locate a point

The same idea is used here. Instead of halving the difference between the points' coordinates, you apply the given ratio.

#### procedure

**1.** | Designate one of your points as point $A$ and the other as point $B$. Let point $A$ have the coordinates $(x_A,y_A)$ and point $B$ have the coordinates $(x_B,y_B)$. |

**2.**
| Suppose the new point splits the line from A to B into the ratio $\alpha:\beta$. This means the new point is $\frac{\alpha}{\alpha+\beta}$ of the line from A. |

**3.** | Find the new point's $x$-coordinate using the formula: $x_A+\frac{\alpha}{\alpha+\beta}(x_B-x_A)$ |

**4.** | Find the new point's $y$-coordinate using the similar formula: $y_A+\frac{\alpha}{\alpha+\beta}(y_B-y_A)$ |

**5.** | You now have that the new point has coordinates $(x_A+\frac{\alpha}{\alpha+\beta}(x_B-x_A),y_A+\frac{\alpha}{\alpha+\beta}(y_B-y_A))$. |

##### Example 1

*Take the line segment with the points A *$(2,-8)$* and B *$(-6,24)$* at either end. The point C is located on this line segment such that it divides the line in the ratio $1:3$. What are the coordinates of point C?*

*The ratio says that point C is a quarter of the distance of the line segment from point A. This is found by dividing the first part of the ratio, *$1$ *by the sum of both parts of the ratio, *$1+3$*:*

$\frac{1}{1+3}=\frac14$

*You want to therefore find the coordinates that are a quarter of the $x$- and $y$-differences from point A.*

*The difference between the $x$ coordinates is *

*$-6-2=-8$*

*To apply the ratio, multiply this by the quarter, then add it to the $2$:*

*$-8\times\frac14=-2$*

*$2+(-2)=0$*

*Thus the $x$-coordinate of point C is $0$. Now do the same for the $y$-coordinate: the difference is*

*$24-(-8)=32$*

*Multiplying this by the quarter (from the ratio) then adding to $-8$ gives*

*$32\times\frac14=8$*

*$-8+8=0$*

*So the $y$-coordinate of point C is also $0$. Hence point C has coordinates $\underline{(0,0)}$.*