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INTRODUCTION
Before 1637, Maths was divided into geometry and algebra.
Equations in geometry and pictures in algebra were not used. Around
1637, a French scientist and philosopher named Ren Descartes
(pronounced "Ray-Nay Day-Cart", 1596-1650) came up with a way to
put these two subjects together. Numbers can be used to accurately
describe the position of any point or coordinate. You may recall
that a system for naming and locating points involves the Cartesian
plane (Figure 7.1). This method was invented in the 17th century by
Descartes.
LEARNING OUTCOMES
TTooppiicc
77 Coordinates
By the end of the topic, you should be able to teach your
students how to:
1. Identify the x-axis, y-axis and the origin on a Cartesian
plane;
2. Determine given coordinates;
3. Find the distance between two points using Pythagoras
Theorem; and
4. Identify the midpoint of the joining two points.
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TOPIC 7 COORDINATES 169
Figure 7.1: Cartesian plane
Pedagogical Content Knowledge To explain Descartes' method,
first ask students to think about using a street map. If they are
trying to find a street that they have never been on before, they
have to look for the street name in the index of the street map.
Suppose the index says that the street is located at D 10. This
means that they have to go across the top of the map and find "D ",
and then go down the side and find "10". Students have to trace
down and across to find the box labelled "D10", and then they have
to look inside the box for the street they need. Somebody figured
out this way to give them directions on the map, by telling them
"how far over" and "how far down" they need to look. Descartes did
something similar. The following subtopic will guide you in
teaching this topic.
INTRODUCTION TO THE CARTESIAN PLANE
The "D10" designation for car parks at a shopping complex is
unambiguous because it is easy for customers to understand the
meaning of D and 10. If the designation is written as "10-D ", the
customers will still know which box of the car park they should go
to, because the "D " would still have been across the top and the
"10" would still have been along the side. But on the CCartesian
plane, both axes are labelled with nnumbers. You can begin a class
by explaining the basic idea of the Cartesian plane by using the
following definitions and terms.
7.1
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7.1.1 Definitions and Terms
The whole flat expanse, top to bottom, side to side, is called
the pplane. When we put the axes on the plane, it is called the
CCartesian ("carr-TEE-zhun") pplane. The name "Cartesian" was
derived after its creator, Descartes.
Coordinates are sets of numbers that describe the pposition of a
location on a surface. The following are the main terminology of
the Cartesian coordinate system:
Coordinates of a point: Represent a pair of number (x, y) on a
plane; x-axis and y-axis: To locate points on a plane, two
perpendicular lines are
used a horizontal line called the x-axis and a vertical line
called the y-axis;
Origin: The intersection point of the x-axis and y-axis;
Coordinate plane: The x-axis, y-axis, and all the points on the
plane; Ordered pairs: Every point on a coordinate plane is named by
a pair of
numbers whose order is important. These pairs of numbers are
written in parentheses and separated by a comma;
x-coordinate: The number to the left of the comma in an ordered
pair is the x-coordinate of the point and indicates the amount of
movement along the x-axis from the origin. The movement is to the
right if the number is positive and to the left if the number is
negative; and
y-coordinate: The number to the right of the comma in an ordered
pair is the y-coordinate of the point and indicates the amount of
movement perpendicular to the x-axis. The movement is above the
x-axis if the number is positive and below the x-axis if the number
is negative.
7.1.2 Axis and Scale
In order to teach this topic effectively, you need to know some
learning aspects about the relationship between axis and scale.
Your students had learned about the basic (counting) number line
back in primary school:
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TOPIC 7 COORDINATES 171
In Form One, they were introduced to zero and nnegatives, which
complete the number line:
Descartes' breakthrough was in taking a second number line,
standing it up on its end, and crossing the first number line at
zero.
The system is based on two straight lines ("axes"),
perpendicular to each other, each of them marked with the distances
from the point where they meet at the origin. The right direction
of the origin on the x-axis and above the origin of the y-axis is
positive, while it is negative on the opposite side.
The number lines, when drawn like this, are called "axes". The
horizontal number line is called the x-axis; the vertical one is
the y-axis (Figures 7.2 and 7.3).
Figure 7.2: 1 unit on the x-axis represents 2 units; 2 units on
the y-axis represent 5 units
Figure 7.3: 2 units on the x-axis represent 5 units; 2 units on
the y-axis represent 3 units
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TOPIC 7 COORDINATES 172
The arrows at the ends of the axes indicate the direction in
which the numbers are getting larger. Therefore, only the axes
should have arrows, and the arrows should be on one end only.
7.1.3 Plotting of Points and Coordinates
Ask your students to take a look at the following: If someone
gave you the direction "(5, 2)" (read as "the point five two" or
just "five two"), where would it be located? To understand the
meaning of "(5, 2)", you have to know the following rule: The
x-coordinate (the number for the x-axis) aalways comes first. The
first number (the first coordinate) is aalways on the horizontal
axis. This is sometimes indicated by referring to points as "(x,
y)" or "x-y points", reinforcing that the first coordinate is
counted off along the x-axis and the second coordinate is counted
off along the y-axis. Some people keep track of this by noting that
the letters are used in alphabetical order. Figure 7.4 illustrates
the quadrants on the Cartesian plane.
Figure 7.4: The quadrants on the Cartesian plane
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TOPIC 7 COORDINATES 173
Example 1: Plot the point (5, 2).
Step 1: Start at the origin, the spot where the axes cross:
Step 2: Move 5 units to the right of the y-axis.
Step 3: Move 2 units above the x-axis. Step 4: Then, draw the
dot.
Finding the location of (5, 2) and drawing the dot is called
"plotting the point (5, 2)". When plotting, remember that the first
number comes from the horizontal axis and the second number comes
from the vertical axis. You always go "so far over" and then "so
far up or down". The following are a couple more examples.
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TOPIC 7 COORDINATES 174
Example 2: Plot the point (4, -5). Step 1: Start at the origin
Step 2: Move 4 units to the right of the
y-axis.
Step 3: Move 5 units below the x-axis.
Step 4: Then, draw the dot.
Note that a negative y-coordinate means that you will be
counting ddown the y-axis, not up.
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TOPIC 7 COORDINATES 175
Example 3: Plot the point (-3, -1) Step 1: Start at the origin
Step 2: Move 3 units to the left of the
y-axis.
Step 3: Move 1 unit below the x-axis. Step 4: Then, draw the
dot.
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TOPIC 7 COORDINATES 176
Give the point in Figure 7.5 that matches each ordered pair
verbally.
(a) (0, -4) (e) (6, 5)(b) (-3, -2) (f) (4, -4)(c) (4, 6) (g) (2,
1)(d) (-4, 0) (h) (-4, 3)
Figure 7.5
SELF-CHECK 7.1
ACTIVITY 7.1
1. Write the coordinates of each of the following points:
2. Rectangle ABCD has coordinates as follows: A(-5,2), B(8,2),
and C(8, -4). Find the coordinates of D.
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TOPIC 7 COORDINATES 177
CARTESIAN PLANE: DISTANCE BETWEEN TWO POINTS, AND MIDPOINTS
In this subtopic, we will revisit the Pythagorean Theorem and
use it to aid us in teaching the Cartesian coordinate system.
7.2.1 Distance between Two Points on the Cartesian Plane
To understand the distance between two points, first your
students must understand the Pythagorean Theorem.
To find the distance between AC, though, simply subtracting is
not sufficient. Triangle ABC is a right-angled triangle with the
hypotenuse AC. Therefore, by applying the Pythagorean Theorem:
2 2 2
2 2
2 2
3 4 5
AC AB BC
AC AB BC
AC
If A is represented by the ordered pair 1 1,x y and C is
represented by the ordered pair 2 2,x y , then AB = 2 1,x x and BC
= 2 1,y y . Then,
222 1 2 1AC x x y y This is stated as a theorem (Refer to
7.2.2).
7.2.2 Use of the Pythagorean Theorem to Find the Distance
between Two Points
Theorem 1: If the coordinates of two points are 1 1,x y and 2
2,x y , then the distance, d, between the two points is given by
the following formula (Distance Formula):
222 1 2 1d x x y y
7.2
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TOPIC 7 COORDINATES 178
Do not let the subscripts scare your students. The subscripts
only indicate that there are a first point and a second point.
Whichever one they call "first" or "second" is up to them. Ask your
students to practise answering all the questions in the following
examples. Then, discuss the solutions with them. Example 6: Use the
Distance Formula to find the distance between the points (3, 4) and
(5, -2). Solution:
1 1 2 2
222 1 2 1
2 2
2 2
Let , 3, 4 and , 5, 2
5 3 2 4
8 6 10
x y x y
d x x y y
Example 7: A triangle has vertices A(12, 5), B(5, 3) and C(12,
1). Show that the triangle is isosceles. Solution: Using the
Distance Formula:
2 2 2 2
2 2 2 2
5 12 3 5 12 5 1 3
7 2 53 7 2 53
AB BC
Since AB = BC, therefore triangle ABC is isosceles. The most
common mistake made by students when using the distance formula is
accidentally mismatching the x-values and y-values. Please remind
your students that they cannot subtract an x from a y, or vice
versa. Make sure they have paired the numbers properly.
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TOPIC 7 COORDINATES 179
Example 8: Refer to the figure in Question 2 of Activity 7.1 to
find the distance of:
(a) AB
(b) BC
Solution:
(a) 8 -513
AB
(b) 2 -46
BC
7.2.3 Coordinates of the Midpoint of Two Points
Midpoint Formula
Point out to your students that numerically, the midpoint of a
segment can be considered as the average of its endpoints. This
concept helps in remembering a formula for finding the midpoint of
a segment given the coordinates of its endpoints. Recall that the
average of two numbers is found by dividing their sum by two.
Theorem 2: If the coordinates of A and B are 1 1,x y and 2 2,x y
, respectively, then the midpoint, M, of AB is given by the
following formula (Midpoint Formula):
1 21 2M ,2 2
y yx x
Example 9: In Figure 7.6, R is the midpoint of Q(-9, -1) and
T(-3, 7). Find its coordinates and use the Distance Formula to
verify that it is in fact the midpoint of QT.
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TOPIC 7 COORDINATES 180
Figure 7.6: Finding the coordinates of the midpoint of a line
segment. Solution: Using the Midpoint Formula:
1 21 2 ,2 2
-9 -3 -1 7,
2 2
-12 6, -6,3
2 2
y yx xR
By applying the Distance Formula,
2 2 2 2
2 2 2 2
6 9 3 1 6 3 3 7
3 4 5 3 4 5
QR TR
Since QR = RT and Q, R, and T are on the same straight line,
therefore R is the midpoint of QT. Example 10: Given that A is (12,
1) and B is (-18, 17), find the midpoint of AB. Solution:
1 1 2 2Let , 12, -1 and , -18,17x y x y
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TOPIC 7 COORDINATES 181
1 21 2Mid point of ,2 2
12 18 -1 17,
2 2-6 16
,2 2
-3,8
y yx xAB
Once your students have mastered the concept well, you can
gradually introduce some exercises on distance and midpoints.
SOLVE PROBLEMS INVOLVING THE DISTANCE BETWEEN TWO POINTS, AND
MIDPOINTS
Choose an appropriate activity to respond to a mathematical
question or represent a situation generated by your students. Help
them to explore the relationship between the Pythagorean Theorem
and the area of a square.
7.3.1 Find the Other Point when the Distance and One Point are
Given
Your students can use the graphical method and the Pythagorean
Theorem to find the coordinates of point Q as in the following
examples.
7.3
Ask your students to solve the following and show their workings
on the whiteboard:
1. Find the distance between the points (-2, -3) and (-4,
4).
2. Find the midpoint of the following pairs of points:
(a) (4, 7) and (8, 10);
(b) (-2, 5) and (3, 17); and
(c) (1, -4) and (7, 2).
ACTIVITY 7.2
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TOPIC 7 COORDINATES 182
Example 11: In Figure 7.7, PQ is a straight line. The
coordinates of P are (-3, 10) and PQ = 11 units. Find the
coordinates of point Q.
Figure 7.7 Solution: Q = (-3 + 11, 10) = (8, 10) since P and Q
are collinear and parallel to the x-axis. Example 12: In Figure
7.8, PQR is an isosceles triangle. If PQ =13units, find the
coordinates of point P.
Figure 7.8
Solution:
2 2
2 213 5 144 12 units.
OP PQ OQ
Thus, P (-12, 0).
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TOPIC 7 COORDINATES 183
7.3.2 Find the Other Point when the Midpoint and One Point are
Given
If given the midpoint and one point, will you be able to find
the coordinates of the other point? The following are some examples
for your students to practise. Example 13: Find the coordinates of
point Q if S(-4, 5) is the midpoint of PQ and the coordinates of P
are (-8, 8). Solution:
2 2 1 1Let , and , -8,8Q x y p x y
1 21 2
22
22
2 2
Mid point of ,2 2
8-8-4, 5 ,
2 28-8
-4 and 52 2, 0,2
y yx xPQ
yx
yx
Q x y
Example 14: Find the value of p so that (-2, 2.5) is the
midpoint of (p, 2) and (-1, 3). Solution: By applying the Midpoint
Formula:
-1 2 3, -2, 2.5
2 2
1 5, -2, 2.5
2 21
-22
1 -4-3
p
p
p
pp
So the answer is p = -3.
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TOPIC 7 COORDINATES 184
7.3.3 Problem Solving Involving Two Points on the Cartesian
Plane
The following are some examples for in-class activity. You can
divide the students into groups or they can work individually.
Example 15: In Figure 7.9, ABCD is a straight line and PQR is a
right triangle. Find the:
(a) Coordinate of D if C is the midpoint of BD.
(b) Coordinate of R.
(c) Midpoint of BP; and
(d) Distance of ST.
Figure 7.9
Solution:
(a) We know that the coordinates of B and C are (-2, 0) and (0,
-1), respectively.
If the coordinates of D are (x, y), then -2 0
0 and -12 2
x y
Hence, x = 2, y = -2. D = (2, -2).
(b) R(2, -0.5)
(c) The midpoint of -2 2 0 1, 0,0.52 2
BP
(d) The distance of ST = 2 (-2) = 4 units.
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TOPIC 7 COORDINATES 185
Example 16: In Figure 7.10, PQRS is a rectangle and M is the
midpoint of PQ. Find the coordinates of point R.
Figure 7.10 Solution: Ask your students to use the graphical
method or the midpoint. Answer: (8, 3) How would you carry out the
activities to strengthen your students knowledge of this topic? You
can ask them to use the algebraic method and graphical method to
solve the following questions.
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TOPIC 7 COORDINATES 186
ACTIVITY 7.3
1. Figure 7.11 shows a Cartesian plane. A is the midpoint of OB
and BC is perpendicular to the x-axis. If the area of OAC is 20
unit2, find the coordinates of C.
Figure 7.11
2. Figure 7.12 shows a Cartesian plane. Given that RSTU is a
parallelogram, find the coordinates of point S.
Figure 7.12
3. The distance between the points A(1, 2k) and B(1 k, 1) is 11
9k . Find the possible values of k.
4. The coordinates of the endpoints of a line segment PQ are
P(3,7) and Q(11,-6). Find the coordinates of the point R such that
PR = QR.
5. Find the perimeter and area of ABC, where vertices are
A(-4,-2), B(8,-2) and C(2,8).
6. Given that M(p, 7) is the midpoint of the line segment
joining the points A(-3,1) and B(11,q), find the values of p and
q.
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TOPIC 7 COORDINATES 187
The ccoordinate plane is a basic concept of coordinate geometry.
It describes a two-dimensional plane in terms of two perpendicular
axes: x and y.
The x-axis indicates the hhorizontal direction while the y-axis
indicates the vertical direction of the plane.
On the coordinate plane, ppoints are indicated by their
positions along the x- and y-axes.
On the coordinate plane, you can use the PPythagorean Theorem to
find the distance between any two points.
If the coordinates of two points are 1 1,x y and 2 2,x y , then
the distance, d, between the two points is given by the following
formula (DDistance Formula):
222 1 2 1d x x y y
To find a point that is halfway between two given points, get
the aaverage of the x-values and the average of the y-values.
If the coordinates of A and B are 1 1,x y and 2 2,x y ,
respectively, the midpoint, M, of AB is given by the following
formula (MMidpoint Formula):
1 21 2M ,2 2
y yx x
Algebraic method
Average
Axis
Cartesian plane
Coordinate geometry
Coordinate plane
Number line
Ordered pairs
Parallelogram
Perimeter
Perpendicular
Points
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TOPIC 7 COORDINATES 188
Coordinates
Distance
Distance formula
Graphical method
Horizontal axis
Hypotenuse
Midpoint
Midpoint formula
Pythagorean Theorem
Quadrant
Scale
Vertical axis
x-axis
x-coordinate
y-axis
y-coordinate
Blair, R. M. (2006).Intermediate algebra. New York, NY: Addison
Wesley.
Cheong, Q. L, & Teh, W. L. (2008). Essential Mathematics
Form 2. Petaling Jaya: Pearson Malaysia.
Lee, L. M. (2007). Mathematics Form 2. Shah Alam: Arah
Pendidikan.
Serge, L. (2008).Basic mathematics. New York, NY: Springer.