# 8.1 Coordinates

8.1 Coordinates

8.1.1 Coordinates
1. The Cartesian coordinate system is a number plane with a horizontal line (x-axis) drawn at right angles to a vertical line (y-axis), intersecting at a point called origin.
2. It is used to locate the position of a point in reference to the x-axis and y-axis.

3. The coordinate of any point are written as an ordered pair (x, y).  The first number is the x-coordinate and the second number is the y-coordinate of the point.

Example:

The coordinates of points A and B are (3, 4) and (–5, –2) respectively.
This means that point A is located 3 units from they-axis and 4 units from the x-axis, whereas point B is located 5 units on the left from the y-axis and 2 units from the x-axis.

4. The coordinate of the origin O is (0, 0).

8.1.2 Scales for the Coordinate Axes
1. The scale for an axis is the number of units represented by a specific length along the axes.

2.
The scale on a coordinate is usually written in the form of a ratio.
Example:
A scale of 1 : 2 means one unit on the graph represents 2 units of the actual length.

3.
Both coordinate axes on the Cartesian plane may have
(a) the same scales, or
(b) different scales.
Example:

1 unit on the x-axis represents 2 units.
1 unit on the y-axis represents 1 unit.
Therefore the scale for x-axis is 1 : 2 and the scale for y-axis is 1 : 1.
Coordinates of:
(4, 3) and Q (10, 5).

8.1.3 Distance between Two Points
1. Finding the distance between two points on a Cartesian plane is the same as finding the length of the straight line joining them.

2. The distance between two points can be calculated by using Pythagoras’ theorem.
Example:
AB = 2 – (–4) = 2 + 4 = 6 units
BC = 5 – (–3) = 5 + 3 = 8 units
By Pythagoras’ theorem,
AC= AB2 + AC2
= 62 + 82
AC = √100
= 10 units
3. Distance is always a positive value.

8.1.4 Midpoint
The midpoint of a straight line joining two points is the middle point that divides the straight line into two equal halves.
$\text{Midpoint,}M=\left(\frac{{x}_{1}+{x}_{2}}{2},\frac{{y}_{1}+{y}_{2}}{2}\right)$

Example
:
The coordinate of the midpoint of (7, –5) and (–3, 11) are
$\begin{array}{l}\left(\frac{7+\left(-3\right)}{2},\frac{-5+11}{2}\right)\\ =\left(\frac{4}{2},\frac{6}{2}\right)\\ =\left(2,3\right)\end{array}$