1. The Cartesian system is a system used to determine the position of a point on a plane by using two number lines which intersect each other at right angles.
2. In the Cartesian system, the horizontal line is called the x-axis and the vertical line is called the y-axis. The intersection point of the x-axis and the y-axis is called the origin.
(B) Plotting points on the Cartesian plane
1. The location of a point on a Cartesian plane is represented by an ordered pair (a, b) known as the coordinates of the point.
2. To plot a point with coordinates (a, b) :
(a) Along the x-axis, starting from the origin, move to the right for the value of positve ‘a’ and to the left for the value of negative ‘a’.
(b) Followed by moving up for the value of positive ‘b’ and down for the value of negative’b’ parallel to the y-axis.
Plot the following points on a Cartesian plane.
A(3, 2) B(- 2, 3) C(- 4, – 3) D(6, – 4)
(C) Stating the coordinates of the points on a Cartesian plane
1. The x-coordinate of a point is the idstance of the point from the y-axis while the y-coordinate of the point is the distance of the point from the x-axis.
2. The Cartesian plane consists of four quadrants. The values of the x-coordinates and y-coordinates in each of the four quadrants of the Cartesian plane are as follows :
3. The coordinates of the origin is (0, 0).
(a) The x-coordinate of any point on the y-axis is always 0(zero).
(b) The y-coordinate of any point on the x-axis is always 0(zero).
The diagram shows a Cartesian plane. State the coordinates of points P, Q and R.
The coordinates of P are (4, 3).
The coordinates of Q are (- 3, 1)
The coordinates of R are (0, -3).
SCALES ON THE COORDINATE AXES
(A) Scales Used on the Coordinate Axes
1 unit on the x-axis represents 2 units.
1 unit on the y-axis represents 2 units.
Therefore, the scale for x and y-axis is 1 : 2.
1 unit on the x-axis represents 5 units.
1 unit on the y-axis represents 3 units.
Therefore, the scale for the x-axis is 1 : 5 and the scale for the y-axis is 1 : 3.
(B) Values on the x-axis and y-axis Based on Given Scales for Both Axes
Mark the values on the x-axis and the y-axis on a Cartesian plane if the scale for the x-axis is 1 : 3 and the scale for the y-axis is 1 : 5.
(C) Stating the Coordinates of a Point Based on Given Scales for Both Axes
State the coordinates of each point marked on the Cartesian plane
P(-5, 20), Q(0, 14), R(8, – 10), S(-7, -12)
(D) Plotting Ponts with Given Coordinates Based on Given Scales
Plot points A(1, 5) and B(-2, 20) on a Cartesian plane if the scale for the x-axis is 1 : 1 and the scale for the y-axis is 1 : 5.
DISTANCE BETWEEN TWO POINTS
The distance between two points is the length of the straight line which joins the two points.
(A) Finding the distance between two points
I. Points with commont y-coordinates
The straight line which joins two points that have the same y-coordinates is parallel to the x-axis. Therefore, the distance between two points, with common y-coordinates is the difference between their x-coordinates.
Find the distance between these points.
(a) A(-3, 5) and B(- 3, 1)
(b) C(1, – 1) and D(1, – 3)
(c) E(4, 3) and F(4, – 2)
(a) Distance between A and B
= difference between the y-coordinates
= 5 – 1
= 4 units
(b) Distance between C and D
= – 1 – (- 3)
= – 1 + 3
= 2 units
(c) Distance between D and F
= 3 – (- 2)
= 3 + 2
= 5 units
(B) Finding the distance between two points using Pythagoras’ theorem
1. The distance between any two points with different x-coordinates and y-coordinates is the length of the straight line joining the two points.
2. The straight line is the hypotenuse of a right-angled triangle whre its two other sides are parallel to the x-axis and y-axis respectively.
For example :
3. We can use the Pythagoras’ theorem to find the distance between these two points. The formula for the distance between two points (a, b) and (c, d) is
Find the distance between point A(2, 2) and point B(10, 8).
(C) Solving problems involving the distance between two points
The diagram show the location of a bank (B), a library (P), Ali’s house (A) and a school (S). Calculate the distance between :
(a) The school and the library
(b) The bank and Ali’s house
(c) Ali’s house and the school
(a) The distance between the school and the library
= length of straight line PS
= 0 – (- 9)
= 9 units
(b) The distance between the bank and Ali’s house
= length of straight line AB
= 10 – (-10)
= 20 units
(c) The distance between Ali’s house and the school is AS
(A) Identifying the Midpoints of Straight Lines
The midpoint of a line joining two points is the point that divides the line into two equal parts.
Therefore, the midpoint for the line AB is (-2, 6).