Coordinates

Coordinate System Basics

The Cartesian plane is a two-dimensional surface defined by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at the origin, which has coordinates $(0,0)$.

Any point on the plane is identified by an ordered pair $(x, y)$, where:

• x is the horizontal distance from the origin (positive to the right, negative to the left)
• y is the vertical distance from the origin (positive upwards, negative downwards)

The plane is divided into four quadrants:

• Quadrant I: $x > 0$, $y > 0$
• Quadrant II: $x < 0$, $y > 0$
• Quadrant III: $x < 0$, $y < 0$
• Quadrant IV: $x > 0$, $y < 0$

Points lying on the axes have either $x=0$ or $y=0$.

For example, the point $(3, -2)$ lies 3 units to the right of the origin and 2 units down, so it is in Quadrant IV.

For instance, the point $(1, 4)$ lies 1 unit right and 4 units up, so it is in Quadrant I.

Reading and Plotting Coordinates

To plot a point $(x, y)$ on the Cartesian plane:

1. Start at the origin $(0,0)$.
2. Move horizontally to the $x$-coordinate (right if positive, left if negative).
3. From there, move vertically to the $y$-coordinate (up if positive, down if negative).
4. Mark the point.

For example, to plot $(-4, 3)$, move 4 units left and 3 units up from the origin.

When reading coordinates from a graph, identify the horizontal position first (the $x$-value), then the vertical position (the $y$-value). Points may have negative coordinates if they lie left of or below the origin.

Interpreting positions means understanding where the point lies in relation to the axes and quadrants. For example, $(0, -5)$ lies on the $y$-axis, 5 units below the origin.

A point with coordinates $(x, 0)$ lies on the $x$-axis, while $(0, y)$ lies on the $y$-axis.

For instance, the point $(-2, 0)$ is 2 units left on the $x$-axis.

Using negative coordinates is essential for fully describing positions on the Cartesian plane.

For example, the point $(-3, -4)$ is 3 units left and 4 units down from the origin, in Quadrant III.

For example, the point $(5, -2)$ lies 5 units right and 2 units down from the origin, so it is in Quadrant IV.

Distance Between Points

The distance between two points on the Cartesian plane can be found by considering the horizontal and vertical distances separately.

If the points share the same $x$-coordinate, the distance is the difference between their $y$-coordinates (vertical distance). Similarly, if they share the same $y$-coordinate, the distance is the difference between their $x$-coordinates (horizontal distance).

For example, the distance between $(3, 7)$ and $(3, 2)$ is:

$\text{Distance} = |7 - 2| = 5\,\mathrm{units}$

If the points have different $x$ and $y$ coordinates, the distance between them is the length of the line segment joining them. At this stage, you can find the horizontal and vertical distances separately, but calculating the exact length using Pythagoras' theorem is covered in another topic on Pythagoras' theorem.

The horizontal distance between points $(x_1, y_1)$ and $(x_2, y_2)$ is $|x_2 - x_1|$.

The vertical distance is $|y_2 - y_1|$.

For example, between $(1, 4)$ and $(5, 9)$:

• Horizontal distance = $|5 - 1| = 4$
• Vertical distance = $|9 - 4| = 5$

For instance, the horizontal distance between $(-2, 3)$ and $(4, 3)$ is $|4 - (-2)| = 6$ units.

Midpoint of a Line Segment

The midpoint of a line segment joining two points is the point exactly halfway between them.

To find the midpoint between points $(x_1, y_1)$ and $(x_2, y_2)$, use the midpoint formula:

$$\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

This means you find the average of the $x$-coordinates and the average of the $y$-coordinates.

For example, the midpoint between $(2, 5)$ and $(6, 9)$ is:

$$\left(\frac{2 + 6}{2}, \frac{5 + 9}{2}\right) = (4, 7)$$

This point $(4, 7)$ lies exactly halfway between the two given points on the line segment.

Midpoints are useful in coordinate geometry to find central points, bisect lines, or check symmetry.