Topic 7 Coordinates

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.

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


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:


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


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


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.


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.


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.


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.


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


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.


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.


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


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


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).


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.


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.


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.


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.


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


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.

Documents

Topic 7 Coordinates