© Boardworks Ltd 2004 1 of 58

KS3 Mathematics

S4 Coordinates and transformations 1

© Boardworks Ltd 2004 2 of 58

Contents

S4 Coordinates and transformations 1

A

A

A

A

AS4.1 Coordinates

S4.5 Rotation symmetry

S4.4 Rotation

S4.2 Reflection

S4.3 Reflection symmetry

© Boardworks Ltd 2004 3 of 58

Coordinates

We can describe the position of any point on a 2-dimensional plane using coordinates.

The coordinate of a point tells us where the point is relative to a starting point or origin.

For example, when we write a coordinate

the first number is called the x-coordinate and the second number is called the y-coordinate.

(3, 5)

x-coordinate

(3, 5)

y-coordinate

(3, 5)

the first number is called the x-coordinate and the second number is called the y-coordinate.

© Boardworks Ltd 2004 4 of 58

Using a coordinate grid

Coordinates are plotted on a grid of squares.

The x-axis and the y-axis intersect at the origin.

The coordinates of the origin are (0, 0).

The lines of the grid are numbered using positive and negative integers as follows.

0 1 2 3 4–4 –3 –2 –1

1

2

3

4

–4

–3

–2

–1

x-axis

y-axis

origin

© Boardworks Ltd 2004 5 of 58

first quadrant

second quadrant

fourth quadrant

third quadrant

0 1 2 3 4–4 –3 –2 –1

1

2

3

4

–4

–3

–2

–1

Quadrants

The coordinate axes divide the grid into four quadrants.

y

x

© Boardworks Ltd 2004 6 of 58

Which quadrant?

© Boardworks Ltd 2004 7 of 58

Coordinates

The first number in the coordinate pair tells you how many units along from the origin the point is in the x-direction.

A positive number means the point is right of the origin and a negative number means it is left.

The second number in the coordinate pair tells you how many units above or below the origin the point is in the y-direction.

A positive number means the point is above the origin and a negative number means it is below.

Remember:

Along the corridor and up (or down) the stairs.Along the corridor and up (or down) the stairs.

© Boardworks Ltd 2004 8 of 58

Plotting points

Plot the point (–3, 5).

0 1 2 3 4 5 6 7–1–2–3–4–5–6–7

1

2

3

4

5

6

7

–2

–4

–6

–3

–5

–7

–1

(–3, 5)

x

y

© Boardworks Ltd 2004 9 of 58

Plotting points

0 1 2 3 4 5 6 7–1–2–3–4–5–6–7

1

2

3

4

5

6

7

–2

–4

–6

–3

–5

–7

–1

(–4, –2)

Plot the point (–4, –2).

x

y

© Boardworks Ltd 2004 10 of 58

0 1 2 3 4 5 6 7–1–2–3–4–5–6–7

1

2

3

4

5

6

7

–2

–4

–6

–3

–5

–7

–1x

y

Plotting points

(6, –7)

Plot the point (6, –7).

© Boardworks Ltd 2004 11 of 58

0 1 2 3 4 5 6 7–1–2–3–4–5–6–7

1

2

3

4

5

6

7

–2

–4

–6

–3

–5

–7

–1x

y

Making quadrilaterals

Where could we add a fourth point to make a parallelogram?

(3, –3)

(–5, –1)

(–5, 4)

(3, 2)

© Boardworks Ltd 2004 12 of 58

Making quadrilaterals

Where could we add a fourth point to make a square?

0 1 2 3 4 5 6 7–1–2–3–4–5–6–7

1

2

3

4

5

6

7

–2

–4

–6

–3

–5

–7

–1x

y

(6, 2)

(2, 6)

(2, –2)

(–2, 2)

© Boardworks Ltd 2004 13 of 58

0 1 2 3 4 5 6 7–1–2–3–4–5–6–7

1

2

3

4

5

6

7

–2

–4

–6

–3

–5

–7

–1x

y

Making quadrilaterals

Where could we add a fourth point to make a rhombus?

(–7, 2) (3, 2)

(–2, 0)

(–2, 4)

© Boardworks Ltd 2004 14 of 58

0 1 2 3 4 5 6 7–1–2–3–4–5–6–7

1

2

3

4

5

6

7

–2

–4

–6

–3

–5

–7

–1x

y

Making quadrilaterals

Where could we add a fourth point to make a kite?

(5, –1)

(2, 2)

(–7, –1)

(2, –4)

© Boardworks Ltd 2004 15 of 58

0 1 2 3 4 5 6 7–1–2–3–4–5–6–7

1

2

3

4

5

6

7

–2

–4

–6

–3

–5

–7

–1x

y

Making quadrilaterals

Where could we add a fourth point to make an arrowhead?

(3, –2)

(3, 3)

(6, 6)(0, 6)

© Boardworks Ltd 2004 16 of 58

0 1 2 3 4 5 6 7–1–2–3–4–5–6–7

1

2

3

4

5

6

7

–2

–4

–6

–3

–5

–7

–1x

y

Making quadrilaterals

Where could we add a fourth point to make a rectangle?

(–3, –3)

(2, 7)

(5, 1)

(–6, 3)

© Boardworks Ltd 2004 17 of 58

Don’t connect three!

© Boardworks Ltd 2004 18 of 58

Finding the mid-point of a horizontal line

Two points A and B have the same y-coordinate.

A is the point (–2, 5) and B is the point (6, 5).

What is the coordinate of the mid-point of the line segment AB?

Let’s call the mid-point M(xm, 5).

xm is the point half-way between –2 and 6.

A(–2, 5) B(6, 5)?M(xm, 5).

?8

© Boardworks Ltd 2004 19 of 58

Finding the mid-point of a horizontal line

Two points A and B have the same y-coordinate.

A is the point (–2, 5) and B is the point (6, 5).

Either, xm = –2 + ½ × 8

A(–2, 5) B(6, 5)?M(xm, 5).

?8

= –2 + 4

= 2

or xm = ½(–2 + 6)

= ½ × 4

= 2

The coordinates of the mid-point of AB are (2, 5).

© Boardworks Ltd 2004 20 of 58

The x-coordinate of the point A is 2The x-coordinate of the point A is 2 and the x-coordinate of the point B is 8.

Finding the mid-point of a line

If A is the point (2, 1) and B is the point (8, 5), what is the mid-point of the line AB?

Start by plotting points A and B on a coordinate grid.

0 1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

B(8, 5)

A(2, 1)

The x-coordinate of the mid-point is half-way between 2 and 8.

2 + 82

= 5x

y

© Boardworks Ltd 2004 21 of 58

and the y-coordinate of the point B is 5.

0 1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

B(8, 5)

A(2, 1)

Finding the mid-point of a line

If A is the point (2, 1) and B is the point (8, 5), what is the mid-point of the line AB?

Start by plotting points A and B on a coordinate grid.

The y-coordinate of the point A is 1

The y-coordinate of the mid-point is half-way between 1 and 5.

1 + 52

= 3

The mid-point of AB is (5, 3).

M(5, 3)

x

y

© Boardworks Ltd 2004 22 of 58

Finding the mid-point of a line

If the coordinates of A are (x1, y1) and the coordinates of B are (x2, y2) then the coordinates of the mid-point of the line segment joining these points are given by:

We can generalize this result to find the mid-point of any line.

x1 + x2

2is the mean of the x-coordinates.

x1 + x2

2,y1 + y2

2

y1 + y2

2is the mean of the y-coordinates.

x1 + x2

2,y1 + y2

2 B(x2, y2)

A(x1, y1)x

y