© Boardworks Ltd 2006 1 of 58 KS3 Mathematics S4 Coordinates and transformations 1.

© Boardworks Ltd 2006 1 of 58

KS3 Mathematics

S4 Coordinates and transformations 1


Contents


A

A

A

A

AS4.1 Coordinates

S4.5 Rotation symmetry

S4.4 Rotation

S4.2 Reflection

S4.3 Reflection symmetry


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.

When we write a coordinate, such as

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.


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.

O 1 2 3 4–4 –3 –2 –1

1

2

3

4

–4

–3

–2

–1

x-axis

y-axis

origin


first quadrant

second quadrant

fourth quadrant

third quadrant

O 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


Which quadrant?


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.


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


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


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).


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)



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)


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


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

(–7, 2) (3, 2)

(–2, 0)

(–2, 4)


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


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

(5, –1)

(2, 2)

(–7, –1)

(2, –4)


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


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

(3, –2)

(3, 3)

(6, 6)(0, 6)


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


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

(–3, –3)

(2, 7)

(5, 1)

(–6, 3)


Don’t connect three!


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


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).


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.

O 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


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

O 1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

B(8, 5)

A(2, 1)


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



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

O


Contents

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S4.2 Reflection


S4.1 Coordinates

S4.4 Rotation




Reflection

An object can be reflected in a mirror line or axis of reflection to produce an image of the object.

Each point in the image must be the same distance from the mirror line as the corresponding point of the original object.


Reflecting shapes

If we reflect the quadrilateral ABCD in a mirror line we label the image quadrilateral A’B’C’D’.

A

B

CD

A’

B’

C’D’

object image

mirror line or axis of reflection

The image is congruent to the original shape.The image is congruent to the original shape.


A

B

CD

A’

B’

C’D’

object image

mirror line or axis of reflection

Reflecting shapes

If we draw a line from any point on the object to its image the line forms a perpendicular bisector to the mirror line.


Reflecting shapes


Reflecting shapes by folding paper

We can make reflections by folding paper.

Draw any polygon at the top of a piece of paper.

Fold the piece of paper back on itself so you can still see the shape.

Place a piece of modeling clay behind the paper and pierce through each vertex of the shape using a compass point.

When the paper is unfolded the vertices of the image will be visible.

Join the vertices together using a ruler.


Reflecting shapes using tracing paper

Suppose we want to reflect this shape in the given mirror line.

Use a piece of tracing paper to carefully trace over the shape and the mirror line with a soft pencil.

When you turn the tracing paper over you will see the following:

Place the tracing paper over the original image making sure the symmetry lines coincide.

Draw around the outline on the back of the tracing paper to trace the image onto the original piece of paper.


Reflect this shape


Reflection on a coordinate grid

The vertices of a triangle lie on the points A(2, 6), B(7, 3) and C(4, –1).

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

A(2, 6)

B(7, 3)

C(4, –1)

Reflect the triangle in the y-axis and label each point on the image.

A’(–2, 6)

B’(–7, 3)

C’(–4, –1)

What do you notice about each point and its image?

x

y



The vertices of a quadrilateral lie on the points A(–4, 6), B(4, 5), C(2, 0) and D(–5, 3).

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

A(–4, 6)B(4, 5)

C(2, 0)

Reflect the quadrilateral in the x-axis and label each point on the image.

A’(–4, –6)B’(4, –5)

D’(–5, –3)


D(–5, 3)

C’(2, 0)x

y



The vertices of a triangle lie on the points A(4, 4), B(7, –1) and C(2, –6).

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

A(4, 4)

C(2, –6)

Reflect the triangle in the line y = x and label each point on the image.

A’(4, 4)

B’(–1, 7)

C’(–6, 2)

x = y


x

y

B(7, –1)


Contents

A

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A




S4.1 Coordinates

S4.4 Rotation

S4.2 Reflection


Reflection symmetry

If you can draw a line through a shape so that one half is a reflection of the other then the shape has reflection or line symmetry.

The mirror line is called a line of symmetry.

one line of symmetry

two lines of symmetry

no lines of symmetry


Reflection symmetry

How many lines of symmetry do the following designs have?

one line of symmetry

five lines of symmetry

three lines of symmetry


Make this shape symmetrical


Planes of symmetry

Is a cube symmetrical?

We can divide the cube into two symmetrical parts here.

This shaded area is called a plane of symmetry.

How many planes of symmetry does a cube have?


Planes of symmetry

We can draw the other eight planes of symmetry for a cube, as follows:


Planes of symmetry

How many planes of symmetry does a cuboid have?

A cuboid has three planes of symmetry.


Planes of symmetry

How many planes of symmetry do the following solids have?

Explain why any right prism will always have at least one plane of symmetry.

An equilateral triangular prism

A square-based pyramid

A cylinder


Investigating shapes made from four cubes


Contents

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A


S4.4 Rotation


S4.1 Coordinates

S4.2 Reflection



Describing a rotation

A rotation occurs when an object is turned around a fixed point.

To describe a rotation we need to know three things:

The angle of rotation.

For example:

½ turn = 180°

The direction of rotation.

Clockwise or anticlockwise?

The centre of rotation.

This is the fixed point about which an object moves.

¼ turn = 90° ¾ turn = 270°


Rotation

Which of the following are examples of rotation in real life?

Can you suggest any other examples?

Walking up stairs?

Riding on a Ferris wheel?

Opening your mouth?

Bending your arm?

Opening a door?

Opening a drawer?


Rotating shapes

If we rotate triangle ABC 90° clockwise about point O the following image is produced:

A

B

CO

A’

B’

C’

A is mapped onto A’, B is mapped onto B’ and C is mapped onto C’.

The image triangle A’B’C’ is congruent to triangle ABC.

90°object

image


Rotating shapes

The centre of rotation can also be inside the shape.

Rotating this shape 90° anticlockwise about point O produces the following image.

90°

O


Determining the direction of a rotation

Sometimes the direction of the rotation is not given.

If this is the case then we use the following rules:

A positive rotation is an anticlockwise rotation.

A negative rotation is an clockwise rotation.

Here are two examples:

A rotation of 60° = an anticlockwise rotation of 60°.

A rotation of –90° = an clockwise rotation of 90°.

Explain why a rotation of 120° is equivalent to a rotation of –240°.


Inverse rotations

The inverse of a rotation maps the image that has been rotated back onto the original object.

90°

For example, the following shape is rotated 90° clockwise about point O.

O

What is the inverse of this rotation?

Either a 90° rotation anticlockwise,

or a 270° rotation clockwise.


Inverse rotations

The inverse of any rotation is either

What is the inverse of a –70° rotation?

Either a 70° rotation,

or a –290° rotation.

A rotation in the same direction, about the same point, but such that the two rotations have a sum of 360°.

A rotation of the same size, about the same point, but in the opposite direction, or


Rotations on a coordinate grid

The vertices of a triangle lie on the points A(2, 6), B(7, 3) and C(4, –1).

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

A(2, 6)

B(7, 3)

C(4, –1)

Rotate the triangle 180° clockwise about the origin and label each point on the image.

A’(–2, –6)

C’(–4, 1)

What do you notice about each point and

its image?

B’(–7, –3)


Rotations on a coordinate grid

The vertices of a triangle lie on the points A(–6, 7), B(2, 4) and C(–4, 4).

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

A(–6, 7)

B(2, 4)

B’(–4, 2) Rotate the triangle 90° anticlockwise about the origin and label each point in the image.

C(–4, 4)

What do you notice about each point and

its image?A’(–7, –6)

C’(–4, –4)


Contents

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S4.1 Coordinates

S4.4 Rotation

S4.2 Reflection



Rotational symmetry

An object has rotational symmetry if it fits exactly onto itself when it is turned about a point at its centre.

The order of rotational symmetry is the number of times the object fits onto itself during a 360° turn.

If the order of rotational symmetry is one, then the object has to be rotated through 360° before it fits onto itself again.

Only objects that have rotational symmetry of two or more are said to have rotational symmetry.

We can find the order of rotational symmetry using tracing paper.


Finding the order of rotational symmetry


Rotational symmetry

Rotational symmetry

order 4

Rotational symmetry

order 3

Rotational symmetry

order 5

What is the order of rotational symmetry for the following designs?


Finding the order of rotational symmetry

This shape has rotational symmetry of order 6 because it maps onto itself in six distinct positions after rotations of 60° about the centre point.

How can you prove that the central shape in this diagram is a regular hexagon?


Symmetry puzzle

© Boardworks Ltd 2006 1 of 58 KS3 Mathematics S4 Coordinates and transformations 1.

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Transcript of © Boardworks Ltd 2006 1 of 58 KS3 Mathematics S4 Coordinates and transformations 1.