Coordinate geometry

A city street grid: every place is fixed by its coordinates.

A point on a graph is described by its coordinates 坐标, sometimes called Cartesian coordinates, written $(x, y)$. The first number is the across value and the second is the up value.

• The two number lines are the axes 坐标轴: the horizontal 水平 $x$-axis and the vertical 竖直 $y$-axis.
• They cross at the origin 原点, the point $(0, 0)$.
• The axes split the grid into four quadrants 象限.

So the point $(3, -2)$ is found by going $3$ to the right and $2$ down.

The axes meet at the origin $O$ and split the plane into four quadrants; $(3,-2)$ means 3 right then 2 down.

Most straight lines can be written as

where $m$ is the gradient 斜率 (steepness) and $c$ is the intercept 截距 — the $y$-value where the line crosses the $y$-axis.

• A line like $x = k$ (for example $x = 3$) is vertical.
• A line like $y = k$ (for example $y = 3$) is horizontal.

A line may also be given as $ax + by = c$. Rearrange it into $y = mx + c$ to read off the gradient and intercept.

Worked example. Find the gradient and $y$-intercept of $5x + 4y = 8$.

So the gradient is $-\tfrac{5}{4}$ and the $y$-intercept is $2$.

A steep mountain road: gradient measures steepness as rise over run.

The gradient measures how steep a line is:

A positive gradient goes up to the right; a negative gradient goes down to the right.

Worked example. Find the gradient of the line through $(1, 2)$ and $(4, 11)$.

For $y=mx+c$, the line crosses the $y$-axis at $c$ and the gradient $m$ is the rise divided by the run.

To draw $y = mx + c$, the quickest way is:

1. Mark the intercept $c$ on the $y$-axis.
2. Use the gradient to step to more points (for $m = 3$, go $1$ right and $3$ up).
3. Join the points with a straight line.

You can also make a small table of values 数值表: choose two or three $x$-values, work out $y$, and plot the points.

Drawing a line fast: mark the intercept, then use the gradient to step to new points

If you know the gradient $m$ and one point on the line, put the point into $y = mx + c$ to find $c$.

Worked example. A line has gradient $3$ and passes through $(1, 2)$. Find its equation.

So the equation is $y = 3x - 1$. (If you are given two points, first find the gradient, then do this.)

A line segment 线段 is the straight piece between two points.

To find the length 长度 of the segment between $(x_1, y_1)$ and $(x_2, y_2)$, use Pythagoras' theorem 勾股定理 on the horizontal and vertical gaps:

To find the midpoint 中点 (the point exactly in the middle), average the coordinates:

Worked example. Find the length and midpoint of the segment from $(1, 2)$ to $(4, 6)$.

The horizontal gap (3) and vertical gap (4) make a right triangle, so the length is $\sqrt{3^2+4^2}=5$; the midpoint is the average of the coordinates.

Parallel 平行 lines never meet, so they have the same gradient.

Worked example. Find the equation of the line parallel to $y = 4x - 1$ that passes through $(1, -3)$.

The gradient is also $4$. Put the point in:

so the line is $y = 4x - 7$.

Parallel lines: same gradient, different intercepts

Two lines are perpendicular 垂直 if they meet at a right angle 直角. Their gradients multiply to $-1$:

In words: flip the fraction and change the sign.

Worked example. Find the gradient of a line perpendicular to $2y = 3x + 1$.

Rearrange: $y = \tfrac{3}{2}x + \tfrac{1}{2}$, so the gradient is $\tfrac{3}{2}$. The perpendicular gradient is $-\tfrac{2}{3}$.

Parallel lines share the same gradient; perpendicular gradients multiply to $-1$.

Perpendicular bisector

The perpendicular bisector 垂直平分线 of a segment cuts it in half at a right angle. To find its equation: get the midpoint, then use the perpendicular gradient through that midpoint.

Worked example. Find the perpendicular bisector of the segment joining $(-3, 8)$ and $(9, -2)$.

• Midpoint: $\left( \frac{-3 + 9}{2}, \frac{8 + (-2)}{2} \right) = (3, 3)$.
• Gradient of the segment: $\frac{-2 - 8}{9 - (-3)} = \frac{-10}{12} = -\tfrac{5}{6}$.
• Perpendicular gradient: $\frac{6}{5}$.

Through $(3, 3)$: $\; 3 = \tfrac{6}{5}(3) + c \Rightarrow c = 3 - \tfrac{18}{5} = -\tfrac{3}{5}$. So the bisector is

• The straight line is $y = mx + c$: $m$ is the gradient and $c$ is where the line crosses the $y$-axis.
• Gradient = (change in $y$) ÷ (change in $x$). Keep the two coordinates in the same order on the top and the bottom.
• Parallel lines have the same gradient; perpendicular lines have gradients that multiply to $-1$ (the negative reciprocal).
• The midpoint is the average of the coordinates; the distance between two points comes from Pythagoras on the differences.