 # Identify Invariant Points

In this worksheet, students will identify invariant points. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Pearson Edexcel, Eduqas

Curriculum topic:   Geometry and Measures

Curriculum subtopic:   Properties and Constructions

Difficulty level:   ### QUESTION 1 of 10

What is an invariant point?

In Transformations, invariant points are points that do not move when you complete a transformations.

When do invariant points occur?

This depends on the transformations.

1) Reflection: Invariant points occur if the mirror line touches the shape.

2) Rotation: Invariant points occur when the centre of rotation is on one of the edges or vertices of the shape to be rotated.

3) Enlargement: Invariant points occur when the centre of enlargement is on one of the edges or vertices of the shape to be rotated.

4) Translations: Invariant points cannot occur on a translation as every point moves the same amount.

Invariant points are points on a shape that...

Invariant points occur in which transformations?

Rotation

Reflection

Translations

Enlargement

The rectangle ABCDhas the coordinates

a = (1,1)

b = (4,1)

c = (1,3)

d = (4,3)

The shape is enlarged by SF 2 around the point (2,1).

Will there be any invariant points?

Yes

No

Not enough information

The rectangle ABCD has the coordinates

a = (1,1)

b = (4,1)

c = (1,3)

d = (4,3)

The shape is enlarged by SF 2 around the point (2,2).

Will there be any invariant points?

Yes

No

Not enough information

A triangle ABC has the coordinates

A = (1,1)

B = (3,1)

C = (1,4)

If the centre of rotation must be an integer point, how many different centres of rotation are there that will give invariant point?

A triangle ABC has the coordinates

A = (1,1)

B = (3,1)

C = (1,4)

For each of the corrdinates below. Select if they would give invariant points if they were used as the centre of rotation.

The rectangle ABCD has the coordinates

a = (1,1)

b = (4,1)

c = (1,3)

d = (4,3)

Which of the following mirror lines will give invariant points?

x = 2

x = 1

x = 5

y = 4

y = 1

The rectangle ABCD has the coordinates

a = (1,1)

b = (4,1)

c = (1,3)

d = (4,3)

This shape is reflected in the mirror line x = 3

How many invariant points are there for the reflection?

The rectangle ABCD has the coordinates

a = (1,1)

b = (4,1)

c = (1,3)

d = (4,3)

This shape is reflected in the mirror line y = 3

Give a integer coordinate that will be invariant.

I translate an object by the vector.

 a b

where a ≠ 0 and b ≠ 0

How many invariant points will I get?

• Question 1

Invariant points are points on a shape that...

EDDIE SAYS
Think of what the word means. Variant - Comes from the root word vary, meaning to change. In- This prefix means NOT InVariant - NOT change
• Question 2

Invariant points occur in which transformations?

Rotation
Reflection
Enlargement
EDDIE SAYS
Invariant points can occur when a different movement applies to each point in the shape. The only translation where everything moves the same amount is the...
• Question 3

The rectangle ABCDhas the coordinates

a = (1,1)

b = (4,1)

c = (1,3)

d = (4,3)

The shape is enlarged by SF 2 around the point (2,1).

Will there be any invariant points?

Yes
EDDIE SAYS
If we draw out the shape and the centre of enlargement, we can see that the centre of enlargement lies on the line AB. This means that this will be an invariant point.
• Question 4

The rectangle ABCD has the coordinates

a = (1,1)

b = (4,1)

c = (1,3)

d = (4,3)

The shape is enlarged by SF 2 around the point (2,2).

Will there be any invariant points?

No
EDDIE SAYS
If we draw out the shape and the centre of enlargement, we can see that the centre of enlargement does not lie on any of the lines This means that there will not be an invariant point.
• Question 5

A triangle ABC has the coordinates

A = (1,1)

B = (3,1)

C = (1,4)

If the centre of rotation must be an integer point, how many different centres of rotation are there that will give invariant point?

6
EDDIE SAYS
In a rotation, the centre must be on one of the lines to give invariant points. Integer points just means that the coordinates must be whole numbers. If you plot this, you will see that there are 6 sets of points that will give invariant points.
• Question 6

A triangle ABC has the coordinates

A = (1,1)

B = (3,1)

C = (1,4)

For each of the corrdinates below. Select if they would give invariant points if they were used as the centre of rotation.

EDDIE SAYS
All you need to be asking yourself here is if the shape are on one of the sides of ABC. If they are, it will give invariant points.
• Question 7

The rectangle ABCD has the coordinates

a = (1,1)

b = (4,1)

c = (1,3)

d = (4,3)

Which of the following mirror lines will give invariant points?

x = 2
x = 1
y = 1
EDDIE SAYS
For a reflection, all that we have to ask is if the mirror line touches the shape at any point. It can be one that goes through the middle of the shape. Not just the ones that go along the sides of the rectangle.
• Question 8

The rectangle ABCD has the coordinates

a = (1,1)

b = (4,1)

c = (1,3)

d = (4,3)

This shape is reflected in the mirror line x = 3

How many invariant points are there for the reflection?

2
EDDIE SAYS
This mirror line goes through the middle of the shape, which means there are only two discrete points that will not move.
• Question 9

The rectangle ABCD has the coordinates

a = (1,1)

b = (4,1)

c = (1,3)

d = (4,3)

This shape is reflected in the mirror line y = 3

Give a integer coordinate that will be invariant.

(1,3)
(2,3)
(3,3)
(4,3)
EDDIE SAYS
The mirror line here goes directly along the line CD. This means every point on this line between x = 1 and x = 4 will be invariant. Because the question asks for integer coordinates, there are only 4 points that you would be allowed to use for this
• Question 10

I translate an object by the vector.

 a b

where a ≠ 0 and b ≠ 0

How many invariant points will I get?

0
EDDIE SAYS
Just testing your understanding. The only way you can get invariant points on a translation is if a and b are equal to 0. A vector of [0,0] would give no movement so can be discounted as an answer.
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