Teach Me Maths

Translations and Column Vectors

Question 11 mark

A shape is translated by the vector \begin{pmatrix}3\\-2\end{pmatrix}

Which of these describes the translation?

Hint

The top number is the movement across. The bottom number is the movement up or down.

Worked solution

A shape is translated by the vector \begin{pmatrix}3\\-2\end{pmatrix}

Which of these describes the translation?

  1. 3 right and 2 up
  2. 2 right and 3 down
  3. 3 right and 2 down
  4. 3 left and 2 down
  1. The top number, 3, is positive, so the shape moves 3 squares to the right.
  2. The bottom number, -2, is negative, so the shape moves 2 squares down.
  3. The translation is 3 right and 2 down.

Question 21 mark

Point A is at (-2,1).

A is translated by the vector \begin{pmatrix}4\\3\end{pmatrix}

Write down the coordinates of the image of A.

Write your answer as (x, y)

Hint

Move 4 squares to the right, then 3 squares up.

Worked solution

Point A is at (-2,1).

A is translated by the vector \begin{pmatrix}4\\3\end{pmatrix}

Write down the coordinates of the image of A.

  1. Moving 4 right changes the x-coordinate: -2+4=2.
  2. Moving 3 up changes the y-coordinate: 1+3=4.
  3. The image of A is (2,4).

Question 31 mark

Point B is at (2,3).

B is translated by the vector \begin{pmatrix}-4\\-5\end{pmatrix}

Write down the coordinates of the image of B.

Write your answer as (x, y)

Hint

A negative top number means move left. A negative bottom number means move down.

Worked solution

Point B is at (2,3).

B is translated by the vector \begin{pmatrix}-4\\-5\end{pmatrix}

Write down the coordinates of the image of B.

  1. Moving 4 left: 2-4=-2.
  2. Moving 5 down: 3-5=-2.
  3. The image of B is (-2,-2).

Question 42 marks

Point P is translated to point Q.

The translation is given by the vector \begin{pmatrix}c\\d\end{pmatrix}

(a)

Find the value of c.

1 mark

(b)

Find the value of d.

1 mark

Hint

Count the squares across from P to Q, then the squares up or down.

Worked solution

Point P is translated to point Q.

The translation is given by the vector \begin{pmatrix}c\\d\end{pmatrix}

Part (a)

Find the value of c.

  1. P has x-coordinate -2 and Q has x-coordinate 3.
  2. From -2 to 3 is 5 squares to the right, so c=5.

Part (b)

Find the value of d.

  1. P has y-coordinate 3 and Q has y-coordinate -1.
  2. From 3 down to -1 is 4 squares down, so d=-4.
  3. The vector is \begin{pmatrix}5\\-4\end{pmatrix}.

Question 52 marks

Triangle ABC is translated by the vector \begin{pmatrix}5\\-3\end{pmatrix} to give triangle A′B′C′.

A′ is shown on the grid.

(a)

Write down the coordinates of B′.

1 mark

Write your answer as (x, y)

(b)

Write down the coordinates of C′.

1 mark

Write your answer as (x, y)

Hint

Move every vertex 5 squares right and 3 squares down.

Worked solution

Triangle ABC is translated by the vector \begin{pmatrix}5\\-3\end{pmatrix} to give triangle A′B′C′.

A′ is shown on the grid.

Part (a)

Write down the coordinates of B′.

  1. B is at (-2,1).
  2. -2+5=3 and 1-3=-2.
  3. B′ is (3,-2).

Part (b)

Write down the coordinates of C′.

  1. C is at (-4,4).
  2. -4+5=1 and 4-3=1.
  3. C′ is (1,1).

Question 61 mark

A shape is translated by the vector \begin{pmatrix}-4\\0\end{pmatrix}

Which of these describes the translation?

Hint

What does a 0 in a column vector tell you about the movement?

Worked solution

A shape is translated by the vector \begin{pmatrix}-4\\0\end{pmatrix}

Which of these describes the translation?

  1. 4 squares down
  2. 4 squares left
  3. 4 squares right
  4. 4 squares left and 4 squares down
  1. The top number, -4, is negative, so the shape moves 4 squares to the left.
  2. The bottom number is 0, so the shape does not move up or down.
  3. The translation is 4 squares left.

Question 71 mark

Point C is at (3,-1).

C is translated by the vector \begin{pmatrix}0\\5\end{pmatrix}

Write down the coordinates of the image of C.

Write your answer as (x, y)

Hint

A top number of 0 means no movement left or right.

Worked solution

Point C is at (3,-1).

C is translated by the vector \begin{pmatrix}0\\5\end{pmatrix}

Write down the coordinates of the image of C.

  1. The top number is 0, so the x-coordinate stays 3.
  2. The bottom number is 5, so move 5 up: -1+5=4.
  3. The image of C is (3,4).

Question 81 mark

A shape is moved 4 squares to the left and 1 square up.

Which column vector describes this translation?

Hint

Decide which number shows the movement across, and whether left is positive or negative.

Worked solution

A shape is moved 4 squares to the left and 1 square up.

Which column vector describes this translation?

  1. \begin{pmatrix}4\\1\end{pmatrix}
  2. \begin{pmatrix}-4\\1\end{pmatrix}
  3. \begin{pmatrix}1\\-4\end{pmatrix}
  4. \begin{pmatrix}-1\\4\end{pmatrix}
  1. The top number is the movement across. Left is negative, so the top number is -4.
  2. The bottom number is the movement up or down. Up is positive, so the bottom number is 1.
  3. The vector is \begin{pmatrix}-4\\1\end{pmatrix}.

Question 92 marks

Shape Q is the image of shape P after a translation by the vector \begin{pmatrix}c\\d\end{pmatrix}

(a)

Find the value of c.

1 mark

(b)

Find the value of d.

1 mark

Hint

Choose one corner of P and count how far it moves to reach the matching corner of Q.

Worked solution

Shape Q is the image of shape P after a translation by the vector \begin{pmatrix}c\\d\end{pmatrix}

Part (a)

Find the value of c.

  1. The bottom left corner of P is at (-5,-4). The bottom left corner of Q is at (1,1).
  2. From x=-5 to x=1 is 6 to the right, so c=6.

Part (b)

Find the value of d.

  1. From y=-4 to y=1 is 5 up, so d=5.
  2. The vector is \begin{pmatrix}6\\5\end{pmatrix}.

Question 103 marks

Triangle PQR is translated by the vector \begin{pmatrix}-6\\-5\end{pmatrix} to give triangle P′Q′R′.

(a)

Write down the coordinates of P′.

1 mark

Write your answer as (x, y)

(b)

Write down the coordinates of Q′.

1 mark

Write your answer as (x, y)

(c)

Write down the coordinates of R′.

1 mark

Write your answer as (x, y)

Hint

Move every vertex 6 squares left and 5 squares down.

Worked solution

Triangle PQR is translated by the vector \begin{pmatrix}-6\\-5\end{pmatrix} to give triangle P′Q′R′.

Part (a)

Write down the coordinates of P′.

  1. P is at (1,2).
  2. 1-6=-5 and 2-5=-3.
  3. P′ is (-5,-3).

Part (b)

Write down the coordinates of Q′.

  1. Q is at (4,2).
  2. 4-6=-2 and 2-5=-3.
  3. Q′ is (-2,-3).

Part (c)

Write down the coordinates of R′.

  1. R is at (1,4).
  2. 1-6=-5 and 4-5=-1.
  3. R′ is (-5,-1).

Question 112 marks

Which of these fully describes the single transformation that maps triangle A onto triangle B?

Hint

Pick one vertex of A. How far across and how far up or down does it move to reach the matching vertex of B?

Worked solution

Which of these fully describes the single transformation that maps triangle A onto triangle B?

  1. Translation by \begin{pmatrix}6\\-3\end{pmatrix}
  2. Translation by \begin{pmatrix}3\\-6\end{pmatrix}
  3. Translation by \begin{pmatrix}-6\\3\end{pmatrix}
  4. Reflection in the y-axis
  1. B is the same size and the same way round as A, so it is a translation.
  2. The right angle of A is at (2,-4). The right angle of B is at (-4,-1).
  3. From x=2 to x=-4 is 6 left, and from y=-4 to y=-1 is 3 up.
  4. The transformation is a translation by \begin{pmatrix}-6\\3\end{pmatrix}.

Question 122 marks

Point A is translated by the vector \begin{pmatrix}3\\-4\end{pmatrix}

The image of A is the point (1,2).

Find the coordinates of A.

Write your answer as (x, y)

Hint

Work backwards from the image: undo each part of the movement.

Worked solution

Point A is translated by the vector \begin{pmatrix}3\\-4\end{pmatrix}

The image of A is the point (1,2).

Find the coordinates of A.

  1. The translation moved A 3 right and 4 down.
  2. To get back to A, move 3 left and 4 up from (1,2).
  3. 1-3=-2 and 2+4=6.
  4. A is at (-2,6).

Question 132 marks

Point P is at (-3,2).

P is translated by the vector \begin{pmatrix}6\\d\end{pmatrix}

The image of P has a y-coordinate of -3.

(a)

Write down the x-coordinate of the image of P.

1 mark

(b)

Find the value of d.

1 mark

Hint

The top number changes the x-coordinate. The bottom number changes the y-coordinate.

Worked solution

Point P is at (-3,2).

P is translated by the vector \begin{pmatrix}6\\d\end{pmatrix}

The image of P has a y-coordinate of -3.

Part (a)

Write down the x-coordinate of the image of P.

  1. The top number is 6, so move 6 right.
  2. -3+6=3

Part (b)

Find the value of d.

  1. The y-coordinate goes from 2 to -3.
  2. That is 5 down, so d=-5.

Question 143 marks

Parallelogram ABCD is translated by the vector \begin{pmatrix}-7\\6\end{pmatrix} to give parallelogram A′B′C′D′.

A′ is shown on the grid.

(a)

Write down the coordinates of B′.

1 mark

Write your answer as (x, y)

(b)

Write down the coordinates of C′.

1 mark

Write your answer as (x, y)

(c)

Write down the coordinates of D′.

1 mark

Write your answer as (x, y)

Hint

Move each vertex 7 squares left and 6 squares up.

Worked solution

Parallelogram ABCD is translated by the vector \begin{pmatrix}-7\\6\end{pmatrix} to give parallelogram A′B′C′D′.

A′ is shown on the grid.

Part (a)

Write down the coordinates of B′.

  1. B is at (4,-5).
  2. 4-7=-3 and -5+6=1.
  3. B′ is (-3,1).

Part (b)

Write down the coordinates of C′.

  1. C is at (5,-3).
  2. 5-7=-2 and -3+6=3.
  3. C′ is (-2,3).

Part (c)

Write down the coordinates of D′.

  1. D is at (3,-2).
  2. 3-7=-4 and -2+6=4.
  3. D′ is (-4,4).

Question 152 marks

Point K is at (2,1).

K is translated by the vector \begin{pmatrix}-3\\5\end{pmatrix}

Jo says the image of K is (7,-2).

(a)

What mistake has Jo made?

1 mark

(b)

Work out the correct coordinates of the image of K.

1 mark

Write your answer as (x, y)

Hint

Work out what Jo did to get 7 and −2.

Worked solution

Point K is at (2,1).

K is translated by the vector \begin{pmatrix}-3\\5\end{pmatrix}

Jo says the image of K is (7,-2).

Part (a)

What mistake has Jo made?

  1. She moved 3 right instead of 3 left
  2. She used the top number for up or down and the bottom number for across
  3. She moved K the wrong way in both directions
  4. She has not made a mistake
  1. Jo worked out 2+5=7 and 1-3=-2.
  2. She used the bottom number for the movement across and the top number for the movement up or down.

Part (b)

Work out the correct coordinates of the image of K.

  1. Move 3 left: 2-3=-1.
  2. Move 5 up: 1+5=6.
  3. The image of K is (-1,6).

Question 16Challenge3 marks

Triangle PQR is translated by the vector \begin{pmatrix}7\\-5\end{pmatrix} to give triangle P′Q′R′.

Triangle P′Q′R′ is shown on the grid. Triangle PQR is not shown.

(a)

Write down the coordinates of P.

1 mark

Write your answer as (x, y)

(b)

Write down the coordinates of Q.

1 mark

Write your answer as (x, y)

(c)

Write down the coordinates of R.

1 mark

Write your answer as (x, y)

Hint

Undo the translation: move each vertex of P′Q′R′ the opposite way.

Worked solution

Triangle PQR is translated by the vector \begin{pmatrix}7\\-5\end{pmatrix} to give triangle P′Q′R′.

Triangle P′Q′R′ is shown on the grid. Triangle PQR is not shown.

Part (a)

Write down the coordinates of P.

  1. The translation moved each point 7 right and 5 down.
  2. To go back, move 7 left and 5 up.
  3. P′ is at (2,-4): 2-7=-5 and -4+5=1.
  4. P is (-5,1).

Part (b)

Write down the coordinates of Q.

  1. Q′ is at (5,-4): 5-7=-2 and -4+5=1.
  2. Q is (-2,1).

Part (c)

Write down the coordinates of R.

  1. R′ is at (5,-1): 5-7=-2 and -1+5=4.
  2. R is (-2,4).

Question 17Challenge4 marks

Triangle PQR is translated by the vector \begin{pmatrix}c\\d\end{pmatrix}

P is translated to the point P′ shown on the grid.

(a)

Find the value of c.

1 mark

(b)

Find the value of d.

1 mark

(c)

Write down the coordinates of R′, the image of R.

2 marks

Write your answer as (x, y)

Hint

Count how far P moves to reach P′. Every other vertex moves in exactly the same way.

Worked solution

Triangle PQR is translated by the vector \begin{pmatrix}c\\d\end{pmatrix}

P is translated to the point P′ shown on the grid.

Part (a)

Find the value of c.

  1. P is at (-5,-3) and P′ is at (1,1).
  2. From x=-5 to x=1 is 6 right, so c=6.

Part (b)

Find the value of d.

  1. From y=-3 to y=1 is 4 up, so d=4.

Part (c)

Write down the coordinates of R′, the image of R.

  1. R is at (-5,-1).
  2. Move 6 right and 4 up: -5+6=1 and -1+4=3.
  3. R′ is (1,3).

Question 18Challenge3 marks

Point R lies on the x-axis.

R is translated by the vector \begin{pmatrix}-4\\2\end{pmatrix}

The image of R lies on the y-axis.

Find the coordinates of R.

Write your answer as (x, y)

Hint

What is the y-coordinate of every point on the x-axis? What is the x-coordinate of every point on the y-axis?

Worked solution

Point R lies on the x-axis.

R is translated by the vector \begin{pmatrix}-4\\2\end{pmatrix}

The image of R lies on the y-axis.

Find the coordinates of R.

  1. R is on the x-axis, so its y-coordinate is 0.
  2. The image is on the y-axis, so its x-coordinate is 0.
  3. The translation moves 4 left, so R must start 4 to the right of 0: its x-coordinate is 4.
  4. R is at (4,0). Check: (4,0) moves to (0,2), which is on the y-axis.

Question 19Challenge4 marks

Quadrilateral ABCD is translated to quadrilateral EFGH.

A moves to E, B moves to F, C moves to G and D moves to H.

A is at (-3,2) and E is at (1,-3).

(a)

B is at (-1,0).

Find the coordinates of F.

2 marks

Write your answer as (x, y)

(b)

H is at (0,1).

Find the coordinates of D.

2 marks

Write your answer as (x, y)

Hint

Work out the movement from A to E first. Every point of the shape moves in the same way.

Worked solution

Quadrilateral ABCD is translated to quadrilateral EFGH.

A moves to E, B moves to F, C moves to G and D moves to H.

A is at (-3,2) and E is at (1,-3).

Part (a)

B is at (-1,0).

Find the coordinates of F.

  1. From A to E: -3 to 1 is 4 right, and 2 to -3 is 5 down.
  2. Move B the same way: -1+4=3 and 0-5=-5.
  3. F is (3,-5).

Part (b)

H is at (0,1).

Find the coordinates of D.

  1. D moves 4 right and 5 down to reach H.
  2. So go back from H: 4 left and 5 up.
  3. 0-4=-4 and 1+5=6.
  4. D is (-4,6).

Question 20Challenge3 marks

Lily was asked to translate triangle PQR by the vector \begin{pmatrix}6\\-6\end{pmatrix}

Her answer, triangle P′Q′R′, is shown on the grid.

One of her vertices is in the wrong place.

(a)

Which vertex is in the wrong place?

1 mark

(b)

Write down the coordinates of where this vertex should be.

2 marks

Write your answer as (x, y)

Hint

Translate each vertex of PQR yourself and compare with Lily's vertices.

Worked solution

Lily was asked to translate triangle PQR by the vector \begin{pmatrix}6\\-6\end{pmatrix}

Her answer, triangle P′Q′R′, is shown on the grid.

One of her vertices is in the wrong place.

Part (a)

Which vertex is in the wrong place?

  1. P′
  2. Q′
  3. R′
  1. P (-5,2) moves to (1,-4), so P′ is correct.
  2. Q (-2,2) moves to (4,-4), so Q′ is correct.
  3. R (-5,5) should move to (1,-1), but Lily's R′ is at (2,-1). R′ is wrong.

Part (b)

Write down the coordinates of where this vertex should be.

  1. R is at (-5,5).
  2. Move 6 right and 6 down: -5+6=1 and 5-6=-1.
  3. R′ should be at (1,-1). Lily moved R 7 to the right instead of 6.

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