**8.1 Coordinates**

**8.1.1 Coordinates**

**1.**The

**Cartesian coordinate system**is a number plane with a horizontal line (

**) drawn at right angles to a vertical line (**

*x*-axis**), intersecting at a point called**

*y-*axis**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 the*y*-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.

**The scale on a coordinate is usually written in the form of a ratio.**

2.

2.

*Example***:**

A scale of 1 : 2 means one unit on the graph represents 2 units of the actual length.

**Both coordinate axes on the Cartesian plane may have**

3.

3.

**(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

**is***x*-axis**1 : 2**and the scale for**is***y*-axis**1 : 1**.Coordinates of:

*P*(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*

^{2 }=

*AB*

^{2}+

*AC*

^{2}

= 6

^{2}+ 8^{2}*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.

$\text{Midpoint,}M=\left(\frac{{x}_{1}+{x}_{2}}{2},\frac{{y}_{1}+{y}_{2}}{2}\right)$

ExampleExample

**:**

The coordinate of the midpoint of (7, –5) and (–3, 11) are

$\begin{array}{l}\left(\frac{7+\left(-3\right)}{2},\frac{-5+11}{2}\right)\\ =\left(\frac{4}{2},\frac{6}{2}\right)\\ =\left(2,3\right)\end{array}$