Question 1
1(a):
Which dashed line above represents the line y=1.
ANSWER: Multiple Choice (Type 1)
A: Line A
B: Line B
C: Line C
D: Line D
Answer: B
Workings:
y=1 is a horizontal line that passes through the y-axis at (0,1).
Marks = 1
1(b):
If Shape A is reflected in the line y=1, what are its invariant points?
ANSWER: Multiple Choice (Type 1)
A: (0,4) and (-1,1)
B: (-1,1)
C: (-4,-2) and (0,-4)
D: (0,-4)
Answer: B
Workings:
The point (-1,1) in Shape A lies on the line of reflection y=1.
When A is reflected in the line y=1, (-1,1) is the only point to stay the same, so it is an invariant point.
Marks = 2
Question 2
A triangle is shown on the grid below.
2(a):
If Shape A is rotated 90\degree clockwise around the point (0,1) to give Shape B, what are the invariant points?
ANSWER: Multiple Choice (Type 1)
A: (0,1)
B: (0,1) and (3,3)
C: (3,3) and (5,1)
D: (5,1)
Answer: A
Workings:
Because the point of rotation is also a vertex on Shape A, this will always be the only invariant point.
Marks = 3
2(b):
If Shape B from part (a) is translated by the vector \begin{pmatrix}1\\5\end{pmatrix} to give Shape C, what are the invariant points between Shape A and Shape C?
ANSWER: Multiple Choice (Type 1)
A: (0,1) and (5,1)
B: (0,1)
C: (5,1)
D: (3,3)
Answer: D
Workings:
Translating Shape B by \begin{pmatrix}1\\5\end{pmatrix} will move the shape 1 unit to the right and 5 units up.
This will give an invariant point at (3,3) where the same vertex from both shapes meet.
Marks = 2
Question 3
3(a):
A triangle is rotated about one of its vertices more than 0° and less than 360°.
How many invariant points does it have?
ANSWER: Multiple Choice (Type 1)
A: 1
B: 2
C: 3
D: 0
Answer: A
Workings:
A triangle rotated about one of its vertices will only have one invariant point when rotated
more than 0\degree and less than 360\degree.
This will be the vertex the triangle is rotated around.
Marks = 1
3(b):
The square above has centre x.
The square is rotated 360° about centre x.
How many invariant vertices does it have?
ANSWER: Multiple Choice (Type 1)
A: 0
B: 1
C: 2
D: 4
Answer: D
Workings:
If a square is rotated 360\degree around its centre, it will complete a full cycle, meaning all points will return to their starting position.
This means all 4 vertices are invariant points.
Marks = 1
Question 4
Shape S is shown on the grid below.
4(a):
Which Shape shows a reflection of S in the line x=1?
ANSWER: Multiple Choice (Type 1)
A: Shape A
B: Shape B
C: Shape C
D: Shape D
Answer: C
Workings:
The points in Shape C are all on the other side of and the same distance from the line x=1 as the corresponding points in Shape S.
Therefore Shape C shows a reflection of Shape S in the line x=1
Marks = 2
4(b):
Describe a translation of your answer to (a) that would result in an invariant point at (4,2) when compared to Shape S.
ANSWER: Multiple Choice (Type 1)
A: \begin{pmatrix}4\\1\end{pmatrix}
B: \begin{pmatrix}6\\0\end{pmatrix}
C: \begin{pmatrix}2\\2\end{pmatrix}
D: \begin{pmatrix}0\\3\end{pmatrix}
Answer: B
Workings:
The corresponding point on Shape C to (4,2) on S is (-2,2).
Therefore to get from (-2,2) to (4,2) requires a translation of \begin{pmatrix}4-(-2)\\2-2\end{pmatrix}=\begin{pmatrix}6\\0\end{pmatrix}.
Marks = 1
Question 5
Triangle R is shown below with vertices at (-2,-3), (-3,1) and (0,1).
Shapes A, B, C and D are also triangles with a common vertex (-2,-3).
5(a):
Which triangle shows an enlargement of R by scale factor 2 about point (-2,-3)?
ANSWER: Multiple Choice (Type 1)
A: Triangle A
B: Triangle B
C: Triangle C
D: Triangle D
Answer: D
Workings:
Draw rays from the centre of enlargement (-2,-3), also a vertex on the triangle, passing through the other two vertices.
These rays sit on the edges of Shape R.
To get from the point (-2,-3) to (-3,1) there is a decrease of 1 horizontally and an increase of 4 vertically.
Because the scale factor is 2, the new shape will decrease by 2 horizontally and increase by 8 vertically, giving a point (-2-2,-3+8) = (-4,5).
To get from the point (-2,-3) to (0,1) there is an increase of 2 horizontally and an increase of 4 vertically.
Because the scale factor is 2, the new shape will increase by 4 horizontally and increase by 8 vertically, giving a point (-2+4,-3+8) = (2,5).
Therefore the new shape must be Shape D.
Marks = 2
5(b):
State the number of invariant points.
ANSWER: Simple Text Answer
Answer: 1
Workings:
Because the bottom vertex is also the centre of enlargement it will always remain the same when enlarging.
Marks = 1
Question 6
Shape A is shown on the grid below.
6(a):
Give the coordinates of all the invariant points if shape A is reflected in the line x=-1.
ANSWER: Multiple Choice (Type 1)
A: (-4,1) and (-2,1)
B: (-5,3) and (-1,3)
C: (-1,3)
D: (-3,5)
Answer: C
Workings:
The point (-1,3) on Shape A lies on the line of reflection x=-1.
Therefore this must be the only invariant point when Shape A is reflected in the line x=-1.
Marks = 1
6(b):
Give the coordinates of all the invariant points if shape A is reflected in the line y=3.
ANSWER: Multiple Choice (Type 1)
A: (-4,1) and (-2,1)
B: (-5,3) and (-1,3)
C: (-3,2)
D: (-3,5)
Answer: B
Workings:
The points (-5,3) and (-1,3) both lie on the line y=3.
Therefore, when Shape A is reflected in the line y=3, the points (-5,3) and (-1,3) will remain in the same place and are
invariant.
Marks = 2
6(c):
Give the coordinates of all the invariant points if shape A is reflected in the line y=-x+2
ANSWER: Multiple Choice (Type 1)
A: There are none
B: (-2,1) and (-1,3)
C: (-4,1) and (-2,1)
D: (-3,5) and (-1,3)
Answer: D
Workings:
The line y=-x+2 passes through the points (-1,3) and (-3,5).
Therefore, if Shape A is reflected in this line, these points will stay in the same place and are invariant points.
Marks = 2
Question 7
7(a):
What are the invariant points of shapes X and Y, reflected in the line y=-x-1?
ANSWER: Multiple Choice (Type 1)
A: (-2,4) & (-5,1)
B: (-5,4)
C: (-3,2) & (-5,4)
D: (-3,2)
Answer: D
Workings:
When Shape X is reflected to give Shape Y there is one point that stays the same, (-3,2).
This is the one and only invariant point.
Marks = 2
7(b):
What are the invariant points of shapes Y and Z given that Shape Y has undergone a 270\degree rotation around the point (-1,0)
ANSWER: Multiple Choice (Type 1)
A: (-1,3) & (-1,-3)
B: (-1,0) & (-4,0)
C: (-1,0)
D: (-4,0)
Answer: C
Workings:
Because Shape Y has undergone a rotation around one of its points, this will be the only invariant point, (-1,0).
Marks = 2
7(c):
Can Z be translated such that there are invariant points on Z?
Give your reasoning.
ANSWER: Multiple Choice (Type 1)
A: Yes, because both X and Y had invariant points in the previous two parts.
B: Yes, because the shape can be moved over any other point.
C: No, because translating will move all the points, so none will stay in the same place.
D: No, because only reflections give invariant points.
Answer: C
Workings:
If a translation occurs, all of the points will move, meaning none of them can remain in place as an invariant point.
Therefore a translation gives no invariant points.
Marks = 1