COORDINATES

COORDINATES

 (A) Identifying the x-axis , y-axis and the origin on a Cartesian plane

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.

Question 1

Plot the following points on a Cartesian plane.

A(3, 2)   B(- 2, 3)   C(- 4, – 3)   D(6, – 4)

Solution

(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).

Question 1

The diagram shows a Cartesian plane.  State the coordinates of points P, Q and R.

 
Solution

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

Question 1

 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.

Solution

(C) Stating the Coordinates of a Point Based on Given Scales for Both Axes

Question 1

State the coordinates of each point marked on the Cartesian plane

Solution

P(-5, 20), Q(0, 14), R(8, – 10), S(-7, -12)

(D) Plotting Ponts with Given Coordinates Based on Given Scales

Question 1

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.

Solution

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.

Question 1

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)

Solution

(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

 

Question 1

Find the distance between point A(2, 2) and point B(10, 8).

Solution

 

(C)   Solving problems involving the distance between two points 

Question 1

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

Solution

(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

 

MIDPOINT

(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.

(B) The Midpoint of a Straight Line Joining Two Points with a Common y-coordinate
In the above diagram, the point (0, 2) is the midpoint of the straight line joining points A(-2, 2) and B(2,2).
Question 1
Find the coordinates of the midpoint of a line joining points A(3, 6) and B(-7, 6)
Solution
y-coordinate for the midpoint = 6
Therefore, the midpoint for the line AB is (-2, 6).
(C) The Midpoint of a Straight Line Joining Two Points with a Common x-coordinate
In the diagram on the right, point (1, 0) is the midpoint of the straight line joining points A(1, 2) and B(1, -2).
Question 1
Find the coordinates of the midpoint of the line joining points P(1, 0) and Q(1, – 4).
Solution
x-coordinate for the midpoint = 1
Therefore, the midpoint for the line PQ is (1, -2).
(D) Coordinates of the Midpoint of a Line Joining Two Points
Question 1
 Find the coordinates of the midpoint of the line joining point M(-1, 5) to point N(3,- 2).
Solution
Question 1
Find the coordinates of the midpoint of the line joining F(8, 2) and G(0, -6).
Solution

2 comments on “COORDINATES

  1. shrinisha says:

    yes maths is fun too

  2. wani says:

    not bad

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