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MATHEMATICS
FORM 4
COORDINATE GEOMETRY
1 | P a g e
Cartesian Coordinates
The position of a point in a plane can be given by an ordered pair of numbers, written as . These are called the Cartesian Coordinates of the point.
NOTE: The coordinate is always stated first followed by the coordinate.
Midpoint of Two Points
The midpoint of any two points, is the half way position of these two point. It the two points
have the coordinates, and , then the midpoint, M, of these two points is given by:
NOTE: The midpoint represents a coordinate on the Cartesian Coordinate system. The
coordinates of the midpoint are the averages of the two -coordinates and of the two -
coordinates of the points.
A (4, 3)
4
3
The coordinates of the
point A are . 4 is
the coordinate and 3
is the coordinate.
MATHEMATICS
FORM 4
COORDINATE GEOMETRY
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Ex. Find the midpoint of:
1.
2.
1. =
=
=
=
2. =
=
=
=
Ex. Find the midpoints of the following:
a.
b.
c.
d.
e.
Always label the values for
before substituting it
into the formula.
MATHEMATICS
FORM 4
COORDINATE GEOMETRY
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Length of A Line/Distance Between Two Points
MATHEMATICS
FORM 4
COORDINATE GEOMETRY
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The length of a line is given by the following formula:
√
Ex. Find the distance between:
A
B To find the length of the line AB, a right
angle triangle can be formed and by
using Pythagoras’ Theorem, the length
of AB can be found.
A (
B (
C (
√
Always label the values for
before substituting it
into the formula.
MATHEMATICS
FORM 4
COORDINATE GEOMETRY
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a.
b.
a. = √
= √
= √
= √
= √
= 13 units
b. = √
= √
= √
= √
= √
8.6 units
Ex. Find the distance between the following pairs of points and give your answer correct
(approximate) to 2 significant figures.
Recall: The sum of two
negative numbers is the
sum of their values and the
sign is kept. The square of
a negative number is
always positive.
MATHEMATICS
FORM 4
COORDINATE GEOMETRY
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a.
b.
c.
d.
e.
Gradient or Slope of A Straight Line
MATHEMATICS
FORM 4
COORDINATE GEOMETRY
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An important concept is the gradient or slope of a line. This is a measure of steepness of a line
relative to the .
The gradient of a line, m, is given by the following formula:
Ex. Find the gradient of the line through and .
=
This line has a positive gradient
since it is sloping upwards.
This line has a negative gradient
since it is sloping downwards.
MATHEMATICS
FORM 4
COORDINATE GEOMETRY
8 | P a g e
=
=
The gradient of this line is
, which is a positive gradient.
Ex. What is the gradient of the line through the points and ?
m =
=
=
The gradient of this line is
, which is a negative gradient.
Ex. State the gradient of the line through the following pairs of points:
a.
b.
c.
MATHEMATICS
FORM 4
COORDINATE GEOMETRY
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d.
e.
Parallel Lines
MATHEMATICS
FORM 4
COORDINATE GEOMETRY
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The lines AB and CD are parallel to each other. Since the slope of both lines are the same, they
both have the same gradient.
Theorem:
Ex. are four points. Which of the lines AB, BC, CA
and DA are parallel?
A
B
C
D
Parallel lines have equal gradients.
Lines with equal gradients are parallel.
MATHEMATICS
FORM 4
COORDINATE GEOMETRY
11 | P a g e
Gradient of AB =
=
=
Gradient of BC =
=
=
Gradient of CA =
=
=
Gradient of DA =
=
=
Perpendicular Lines
MATHEMATICS
FORM 4
COORDINATE GEOMETRY
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Two lines are perpendicular when the meet each other at right angles, 90o as shown in the
diagram below.
Theorem:
NOTE: If the gradient of one line is given and it is known that another line is perpendicular to it,
to find the gradient of the perpendicular line, find the negative reciprocal of the known gradient.
Ex. Find the negative reciprocal of the following numbers:
A
B
C
D
Let the gradient of AB be
and let the gradient of CD be
. There exist a relationship
such that the gradients can
determine if two lines are
perpendicular.
If are the gradients of two lines, then the product
of their gradients ( is
MATHEMATICS
FORM 4
COORDINATE GEOMETRY
13 | P a g e
Ex. The points A, B and C have the coordinates respectively.
Show that AB is perpendicular to BC.
To show AB is perpendicular to BC, find the gradients of both AB and BC and see if they are the
negative reciprocals of each other.
Gradient of AB =
=
=
Gradient of BC =
=
=
Ex. CD is the perpendicular bisector of the line joining and . State the
coordinates of the point where CD intersect AB and the gradient of CD.
Number Negative Reciprocal
The reciprocal of a number is
interchanging the numerator with the
denominator or flipping the fraction.
To find the negative of a number,
multiply it by .
MATHEMATICS
FORM 4
COORDINATE GEOMETRY
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The perpendicular bisector CD meets the line AB at the mid-point of AB.
M =
=
=
=
The gradient therefore of the perpendicular bisector is :
Equation of a Straight Line
A straight line on a graph is obtained from an equation by selecting suitable values and
substituting them into the equation to get corresponding values for each value. The general
form of a straight line is:
MATHEMATICS
FORM 4
COORDINATE GEOMETRY
15 | P a g e
Where are variables, m is the gradient and c is the intercept on the .
However the equation of a straight line can be obtained when coordinates are given. There are 2
ways of obtaining the equation of a straight line:
1. Given one point and the gradient
2. Given two points
1. One-Point, Gradient Form
When one point and the gradient is given, the following formula is used to obtain the equation of
a straight line.
Ex. What is the equation of the line through with gradient
?
=
=
=
=
=
2. Given Two Points
When two points are given, the gradient is found first and by using this gradient and any of the
two points given, the equation of the straight line can be found.
MATHEMATICS
FORM 4
COORDINATE GEOMETRY
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Ex. Find the equation of the line through and
First find the gradient:
m =
m =
m =
=
=
=
=
=
=
=
Equations of Parallel and Perpendicular Lines
Ex. Find the equations of the lines through the point which are parallel and perpendicular
to the line .
MATHEMATICS
FORM 4
COORDINATE GEOMETRY
17 | P a g e
First write the equation in the general form
The gradient of this line is :
Using , the point and the gradient ,the equation of the line parallel to
is :
The gradient of the perpendicular line is :
Using , the point and the gradient ,the equation of the line
perpendicular to is :
May/June 2014
MATHEMATICS
FORM 4
COORDINATE GEOMETRY
18 | P a g e
MATHEMATICS
FORM 4
COORDINATE GEOMETRY
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January 2014
MATHEMATICS
FORM 4
COORDINATE GEOMETRY
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MATHEMATICS
FORM 4
COORDINATE GEOMETRY
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May/June 2013
and are the end points of a line segment AB. Determine:
a. The gradient of AB [2 marks]
b. The coordinates of the midpoint of AB [2 marks]
c. The equation of the perpendicular bisector of AB [3 marks]
MATHEMATICS
FORM 4
COORDINATE GEOMETRY
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May/June 2012
The line passes through the points and . Determine:
a. The gradient of the line, [2 marks]
b. The equation of the line, [2 marks]
c. The midpoint of the line segment, TS [1 mark]
d. The length of the line segment, TS [2 marks]
