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Exam questions · Maths

Transformations and Similarity

  • 30 exam questions
  • 87 marks
  • 45 quick checks

Reflections and Translations

Just this lesson
  1. 1 Write down [2 marks]

    Write down the coordinates of the image of the point \((3, 4)\) after a reflection in the \(y\)-axis.

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    Model answer

    A reflection in the \(y\)-axis changes the sign of the \(x\)-coordinate, so the image is \((-3, 4)\).

    Mark scheme

    • The \(x\)-coordinate changes sign — M1
    • \((-3, 4)\) — A1
  2. 2 Reflect [4 marks]

    Triangle \(P\) is drawn on the grid. (a) Reflect triangle \(P\) in the line \(x = 4\). Label the image \(Q\). (2 marks) (b) Translate triangle \(P\) by the vector \(\begin{pmatrix} 2 \\ 3 \end{pmatrix}\). Label the image \(R\). (2 marks)

    Triangle P on a grid with the mirror line x equals 4.
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    Model answer

    (a) The vertices of \(P\) are \((1, 1)\), \((3, 1)\) and \((1, 4)\), which are 3, 1 and 3 units to the left of \(x = 4\). So \(Q\) has vertices \((7, 1)\), \((5, 1)\) and \((7, 4)\). (b) Adding 2 to each \(x\) and 3 to each \(y\) gives \(R\) with vertices \((3, 4)\), \((5, 4)\) and \((3, 7)\).

    Mark scheme

    • (a) Reflects at least two vertices correctly — M1
    • (a) \(Q\) with vertices \((7, 1)\), \((5, 1)\) and \((7, 4)\) — A1
    • (b) Translates at least two vertices correctly — M1
    • (b) \(R\) with vertices \((3, 4)\), \((5, 4)\) and \((3, 7)\) — A1
  3. 3 Describe [3 marks]

    Triangle \(B\) is the image of triangle \(A\) after a single transformation. (a) Describe fully the single transformation that maps \(A\) onto \(B\). (2 marks) (b) Triangle \(A\) has an area of 3 square units. Write down the area of triangle \(B\). (1 mark)

    Triangle A and its image triangle B on a grid.
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    Model answer

    (a) Corresponding points are the same distance from the line \(y = 2\) on opposite sides, so it is a reflection in the line \(y = 2\). (b) A reflection does not change the size, so the area of \(B\) is also 3 square units.

    Mark scheme

    • (a) Reflection — B1
    • (a) In the line \(y = 2\) — B1
    • (b) 3 — B1
  4. 4 Write down [3 marks]

    \(A\) is the point \((2, 3)\). \(A\) is translated by the vector \(\begin{pmatrix} -4 \\ 1 \end{pmatrix}\) to give the point \(B\). \(B\) is reflected in the \(x\)-axis to give the point \(C\). Write down the coordinates of (a) \(B\), (b) \(C\). (3 marks)

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    Model answer

    (a) \(B = (2 - 4, 3 + 1) = (-2, 4)\). (b) A reflection in the \(x\)-axis changes the sign of the \(y\)-coordinate, so \(C = (-2, -4)\).

    Mark scheme

    • (a) \((-2, 4)\) — B1
    • (b) Changes the sign of the \(y\)-coordinate of their \(B\) — M1
    • (b) \((-2, -4)\) — A1 (follow through from (a))
  5. 5 Work out [3 marks]

    A translation maps the point \((-1, 4)\) onto the point \((5, -2)\). (a) Write down the column vector of the translation. (2 marks) (b) The same translation maps \((3, 3)\) onto the point \(Q\). Write down the coordinates of \(Q\). (1 mark)

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    Model answer

    (a) The change in \(x\) is \(5 - (-1) = 6\) and the change in \(y\) is \(-2 - 4 = -6\), so the vector is \(\begin{pmatrix} 6 \\ -6 \end{pmatrix}\). (b) \(Q = (3 + 6, 3 - 6) = (9, -3)\).

    Mark scheme

    • (a) \(5 - (-1)\) or \(-2 - 4\) — M1
    • (a) \(\begin{pmatrix} 6 \\ -6 \end{pmatrix}\) — A1
    • (b) \((9, -3)\) — B1 (follow through from (a))
  6. 6 Work out [3 marks]

    The point \(P\) is \((3, 1)\). \(P\) is reflected in the line \(y = x\) to give \(Q\). \(Q\) is reflected in the \(y\)-axis to give \(R\). Write down the coordinates of \(R\).

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    Model answer

    Reflecting in \(y = x\) swaps the coordinates, so \(Q = (1, 3)\). Reflecting in the \(y\)-axis changes the sign of \(x\), so \(R = (-1, 3)\).

    Mark scheme

    • \(Q = (1, 3)\) — B1
    • Changes the sign of the \(x\)-coordinate of their \(Q\) — M1
    • \((-1, 3)\) — A1

Quick check

  1. 1

    What is the image of the point \((3, 5)\) in the \(x\)-axis?

    1. A\((-3, 5)\)
    2. B\((3, -5)\)
    3. C\((5, 3)\)
    4. D\((-3, -5)\)
    Show answerHide answer

    B: \((3, -5)\)

    A reflection in the \(x\)-axis changes the sign of the \(y\)-coordinate.

  2. 2

    What is the image of the point \((2, 3)\) in the \(y\)-axis?

    1. A\((-2, 3)\)
    2. B\((2, -3)\)
    3. C\((3, 2)\)
    4. D\((-2, -3)\)
    Show answerHide answer

    A: \((-2, 3)\)

    A reflection in the \(y\)-axis changes the sign of the \(x\)-coordinate.

  3. 3

    What is the image of \((4, 1)\) in the line \(y = x\)?

    1. A\((-1, -4)\)
    2. B\((4, -1)\)
    3. C\((-4, 1)\)
    4. D\((1, 4)\)
    Show answerHide answer

    D: \((1, 4)\)

    In the line \(y = x\) the coordinates swap.

  4. 4

    The point \((5, 2)\) is reflected in the line \(x = 3\). What is the image?

    1. A\((-5, 2)\)
    2. B\((3, 2)\)
    3. C\((1, 2)\)
    4. D\((1, -2)\)
    Show answerHide answer

    C: \((1, 2)\)

    \((5, 2)\) is 2 right of the line, so the image is 2 left of it, at \((1, 2)\).

  5. 5

    What does the column vector \(\begin{pmatrix} -2 \\ 5 \end{pmatrix}\) mean?

    1. A2 right and 5 up
    2. B2 left and 5 up
    3. C5 left and 2 up
    4. D2 left and 5 down
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    B: 2 left and 5 up

    The top number is left or right, and the bottom number is up or down.

  6. 6

    The point \((4, 1)\) is translated by \(\begin{pmatrix} -3 \\ 2 \end{pmatrix}\). What is the image?

    1. A\((1, 3)\)
    2. B\((7, 3)\)
    3. C\((1, -1)\)
    4. D\((7, -1)\)
    Show answerHide answer

    A: \((1, 3)\)

    \((4 - 3, 1 + 2) = (1, 3)\).

  7. 7

    A translation takes \((2, 5)\) to \((7, 3)\). What is the column vector?

    1. A\(\begin{pmatrix} -5 \\ 2 \end{pmatrix}\)
    2. B\(\begin{pmatrix} 5 \\ 2 \end{pmatrix}\)
    3. C\(\begin{pmatrix} 9 \\ 8 \end{pmatrix}\)
    4. D\(\begin{pmatrix} 5 \\ -2 \end{pmatrix}\)
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    D: \(\begin{pmatrix} 5 \\ -2 \end{pmatrix}\)

    The change in \(x\) is \(7 - 2 = 5\) and the change in \(y\) is \(3 - 5 = -2\).

  8. 8

    What must you give to describe a reflection fully?

    1. AThe scale factor
    2. BThe centre and the angle
    3. CThe equation of the mirror line
    4. DA column vector
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    C: The equation of the mirror line

    A reflection is described by its mirror line, such as \(x = 1\).

  9. 9

    What is the image of \((4, 1)\) in the line \(y = -x\)?

    1. A\((1, 4)\)
    2. B\((-1, -4)\)
    3. C\((-4, -1)\)
    4. D\((-1, 4)\)
    Show answerHide answer

    B: \((-1, -4)\)

    In the line \(y = -x\) the coordinates swap and both change sign.

Rotations

Just this lesson
  1. 1 Write down [2 marks]

    Write down the coordinates of the image of the point \((3, -2)\) after a rotation of \(180^\circ\) about the origin.

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    Model answer

    A \(180^\circ\) rotation about the origin changes the sign of both coordinates, so the image is \((-3, 2)\).

    Mark scheme

    • Both coordinates change sign — M1
    • \((-3, 2)\) — A1
  2. 2 Rotate [3 marks]

    Rotate triangle \(T\) through \(90^\circ\) clockwise about the point \(O\), the origin. (3 marks)

    Triangle T on a grid with the origin O marked.
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    Model answer

    A \(90^\circ\) clockwise turn about the origin sends \((x, y)\) to \((y, -x)\). The vertices \((1, 1)\), \((3, 1)\) and \((1, 4)\) go to \((1, -1)\), \((1, -3)\) and \((4, -1)\).

    Mark scheme

    • Rotates at least two vertices by \(90^\circ\) about the origin — M1
    • At least two vertices correct, such as \((1, -1)\) and \((1, -3)\) — A1
    • Triangle with vertices \((1, -1)\), \((1, -3)\) and \((4, -1)\) — A1
  3. 3 Describe [3 marks]

    Describe fully the single transformation that maps triangle \(A\) onto triangle \(B\).

    Triangle A and its image triangle B on a grid.
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    Model answer

    The point \((2, 1)\) goes to \((-1, 2)\), and \((5, 1)\) goes to \((-1, 5)\). This is the rule \((x, y) \to (-y, x)\), which is a rotation of \(90^\circ\) anticlockwise about the origin.

    Mark scheme

    • Rotation — B1
    • \(90^\circ\) anticlockwise — B1
    • About the origin, or the point (0, 0) — B1
  4. 4 Find [2 marks]

    A rotation of \(180^\circ\) maps the point \((1, 4)\) onto the point \((5, 0)\). Find the coordinates of the centre of the rotation.

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    Model answer

    The centre of a \(180^\circ\) rotation is the midpoint of a point and its image: \(\left(\dfrac{1 + 5}{2}, \dfrac{4 + 0}{2}\right) = (3, 2)\).

    Mark scheme

    • \(\dfrac{1 + 5}{2}\) or \(\dfrac{4 + 0}{2}\) — M1
    • \((3, 2)\) — A1
  5. 5 Work out [3 marks]

    The point \(P\) is \((4, 1)\). \(P\) is rotated through \(90^\circ\) clockwise about the origin to give \(Q\). \(Q\) is reflected in the \(x\)-axis to give \(R\). Write down the coordinates of \(R\).

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    Model answer

    The rotation sends \((x, y)\) to \((y, -x)\), so \(Q = (1, -4)\). Reflecting in the \(x\)-axis changes the sign of \(y\), so \(R = (1, 4)\).

    Mark scheme

    • \(Q = (1, -4)\) — B1
    • Changes the sign of the \(y\)-coordinate of their \(Q\) — M1
    • \((1, 4)\) — A1
  6. 6 Work out [3 marks]

    The point \((3, 2)\) is rotated through \(90^\circ\) clockwise about the point \((1, 0)\). Work out the coordinates of the image.

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    Model answer

    The point is 2 right and 2 up from the centre. A \(90^\circ\) clockwise turn sends \((x, y)\) to \((y, -x)\), so the new position is 2 right and 2 down from the centre. The image is \((1 + 2, 0 - 2) = (3, -2)\).

    Mark scheme

    • Position relative to the centre, \((2, 2)\) — M1
    • Rotates it to \((2, -2)\) relative to the centre — M1
    • \((3, -2)\) — A1

Quick check

  1. 1

    What three details describe a rotation?

    1. AThe mirror line and the angle
    2. BThe scale factor and the centre
    3. CThe centre, the angle and the direction
    4. DThe column vector and the angle
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    C: The centre, the angle and the direction

    A rotation is described by its centre, its angle and its direction.

  2. 2

    What is the image of \((3, -2)\) in a \(180^\circ\) rotation about the origin?

    1. A\((3, 2)\)
    2. B\((-3, 2)\)
    3. C\((2, -3)\)
    4. D\((-3, -2)\)
    Show answerHide answer

    B: \((-3, 2)\)

    A \(180^\circ\) turn about the origin changes the sign of both coordinates.

  3. 3

    What is the image of \((2, 5)\) in a \(90^\circ\) anticlockwise rotation about the origin?

    1. A\((-5, 2)\)
    2. B\((5, -2)\)
    3. C\((-2, -5)\)
    4. D\((5, 2)\)
    Show answerHide answer

    A: \((-5, 2)\)

    A \(90^\circ\) anticlockwise turn sends \((x, y)\) to \((-y, x)\).

  4. 4

    What is the image of \((3, 1)\) in a \(90^\circ\) clockwise rotation about the origin?

    1. A\((-1, 3)\)
    2. B\((-3, -1)\)
    3. C\((-1, -3)\)
    4. D\((1, -3)\)
    Show answerHide answer

    D: \((1, -3)\)

    A \(90^\circ\) clockwise turn sends \((x, y)\) to \((y, -x)\).

  5. 5

    Why is no direction needed to describe a \(180^\circ\) rotation?

    1. AThe shape does not move
    2. BThe direction is always clockwise
    3. CA half turn is the same in both directions
    4. DThe centre decides the direction
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    C: A half turn is the same in both directions

    A half turn clockwise ends in the same place as a half turn anticlockwise.

  6. 6

    Is the image of a rotation congruent to the object?

    1. ANo, it is always larger
    2. BYes, it has the same size and shape
    3. CNo, it is always smaller
    4. DOnly for a half turn
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    B: Yes, it has the same size and shape

    A rotation does not change lengths or angles.

  7. 7

    What is the image of \((-2, 4)\) in a \(90^\circ\) clockwise rotation about the origin?

    1. A\((4, 2)\)
    2. B\((-4, -2)\)
    3. C\((2, 4)\)
    4. D\((-4, 2)\)
    Show answerHide answer

    A: \((4, 2)\)

    \((x, y)\) goes to \((y, -x)\), so \((-2, 4)\) goes to \((4, 2)\).

  8. 8

    The point \((4, 3)\) is rotated through \(180^\circ\) about the point \((1, 1)\). What is the image?

    1. A\((-4, -3)\)
    2. B\((-3, -2)\)
    3. C\((2, 1)\)
    4. D\((-2, -1)\)
    Show answerHide answer

    D: \((-2, -1)\)

    The point is 3 right and 2 up from the centre, so the image is 3 left and 2 down: \((1 - 3, 1 - 2) = (-2, -1)\).

  9. 9

    A \(180^\circ\) rotation takes \((1, 5)\) to \((5, 1)\). What is the centre of rotation?

    1. A\((0, 0)\)
    2. B\((4, 4)\)
    3. C\((3, 3)\)
    4. D\((6, 6)\)
    Show answerHide answer

    C: \((3, 3)\)

    The centre is the midpoint of a point and its image: \(\left(\dfrac{1 + 5}{2}, \dfrac{5 + 1}{2}\right) = (3, 3)\).

Enlargements

Just this lesson
  1. 1 Write down [2 marks]

    A rectangle is 3 cm by 5 cm. It is enlarged by a scale factor of 4. Write down the length and the width of the enlarged rectangle.

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    Model answer

    Each length is multiplied by 4, so the enlarged rectangle is \(3 \times 4 = 12\) cm by \(5 \times 4 = 20\) cm.

    Mark scheme

    • \(3 \times 4\) or \(5 \times 4\) — M1
    • 12 cm by 20 cm — A1
  2. 2 Enlarge [3 marks]

    Enlarge triangle \(T\) by a scale factor of 3 with the centre of enlargement \(O\), the origin. (3 marks)

    Triangle T on a grid with the centre of enlargement O at the origin.
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    Model answer

    Multiply each coordinate by 3: \((1, 1)\) goes to \((3, 3)\), \((3, 1)\) goes to \((9, 3)\) and \((1, 2)\) goes to \((3, 6)\).

    Mark scheme

    • Enlarges at least two vertices by a scale factor of 3 — M1
    • At least two vertices correct — A1
    • Triangle with vertices \((3, 3)\), \((9, 3)\) and \((3, 6)\) — A1
  3. 3 Describe [3 marks]

    Describe fully the single transformation that maps triangle \(A\) onto triangle \(B\).

    Triangle A and its image triangle B on a grid.
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    Model answer

    The bottom of \(A\) is 2 long and the bottom of \(B\) is 4 long, so the scale factor is 2. The lines through matching corners, such as \((2, 1)\) and \((3, 2)\), and \((4, 1)\) and \((7, 2)\), meet at \((1, 0)\). So it is an enlargement, scale factor 2, centre \((1, 0)\).

    Mark scheme

    • Enlargement — B1
    • Scale factor 2 — B1
    • Centre \((1, 0)\) — B1
  4. 4 Work out [3 marks]

    A triangle has sides of 4 cm, 6 cm and 8 cm. It is enlarged by a scale factor of 1.5. (a) Work out the lengths of the sides of the enlarged triangle. (2 marks) (b) Work out the perimeter of the enlarged triangle. (1 mark)

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    Model answer

    (a) \(4 \times 1.5 = 6\), \(6 \times 1.5 = 9\) and \(8 \times 1.5 = 12\). (b) \(6 + 9 + 12 = 27\) cm.

    Mark scheme

    • (a) At least one length multiplied by 1.5 — M1
    • (a) 6 cm, 9 cm and 12 cm — A1
    • (b) 27 cm — B1 (follow through from (a))
  5. 5 Work out [3 marks]

    A square has sides of 12 cm. It is enlarged by a scale factor of \(\dfrac{1}{3}\). (a) Write down the side length of the enlarged square. (1 mark) (b) Work out the area of the enlarged square. (2 marks)

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    Model answer

    (a) \(12 \times \dfrac{1}{3} = 4\) cm. (b) \(4 \times 4 = 16\) cm\(^2\).

    Mark scheme

    • (a) 4 cm — B1
    • (b) \(4 \times 4\) — M1
    • (b) 16 cm\(^2\) — A1 (follow through from (a))
  6. 6 Work out [3 marks]

    The point \(A\) is \((3, 2)\). \(A\) is enlarged by a scale factor of \(-2\) with the centre \((1, 1)\). Work out the coordinates of the image of \(A\).

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    Model answer

    \(A\) is 2 right and 1 up from the centre. With a scale factor of \(-2\), the image is 4 left and 2 down from the centre, at \((1 - 4, 1 - 2) = (-3, -1)\).

    Mark scheme

    • Position relative to the centre, \((2, 1)\) — M1
    • \((-4, -2)\) relative to the centre — M1
    • \((-3, -1)\) — A1

Quick check

  1. 1

    A side of 3 cm is enlarged to 12 cm. What is the scale factor?

    1. A\(9\)
    2. B\(\dfrac{1}{4}\)
    3. C\(36\)
    4. D\(4\)
    Show answerHide answer

    D: \(4\)

    \(\dfrac{12}{3} = 4\).

  2. 2

    The point \((2, 3)\) is enlarged by scale factor 3 with centre \((0, 0)\). What is the image?

    1. A\((5, 6)\)
    2. B\((6, 3)\)
    3. C\((6, 9)\)
    4. D\((9, 6)\)
    Show answerHide answer

    C: \((6, 9)\)

    Multiply both coordinates by 3.

  3. 3

    What happens to the angles in an enlargement?

    1. AThey are multiplied by the scale factor
    2. BThey stay the same
    3. CThey are halved
    4. DThey are added to the scale factor
    Show answerHide answer

    B: They stay the same

    An enlargement keeps the angles, so the shapes are similar.

  4. 4

    What does an enlargement with scale factor \(\dfrac{1}{2}\) do to a shape?

    1. AIt halves every length
    2. BIt doubles every length
    3. CIt halves every angle
    4. DIt leaves the shape unchanged
    Show answerHide answer

    A: It halves every length

    A fractional scale factor less than 1 makes the shape smaller.

  5. 5

    The centre of enlargement is \((1, 2)\) and the scale factor is 3. What is the image of \((3, 3)\)?

    1. A\((9, 9)\)
    2. B\((5, 4)\)
    3. C\((6, 5)\)
    4. D\((7, 5)\)
    Show answerHide answer

    D: \((7, 5)\)

    \((3, 3)\) is 2 right and 1 up from the centre, so the image is 6 right and 3 up: \((7, 5)\).

  6. 6

    What must you give to describe an enlargement fully?

    1. AThe mirror line
    2. BThe angle and the direction
    3. CThe scale factor and the centre
    4. DA column vector
    Show answerHide answer

    C: The scale factor and the centre

    An enlargement is described by its scale factor and its centre.

  7. 7

    A triangle with sides 4, 6 and 8 is enlarged to give a triangle with sides 10, 15 and 20. What is the scale factor?

    1. A\(2\)
    2. B\(2.5\)
    3. C\(6\)
    4. D\(3\)
    Show answerHide answer

    B: \(2.5\)

    \(\dfrac{10}{4} = 2.5\).

  8. 8

    Which transformation is the same as an enlargement with scale factor \(-1\)?

    1. AA rotation of \(180^\circ\) about the centre
    2. BA reflection in a line through the centre
    3. CA translation
    4. DA rotation of \(90^\circ\)
    Show answerHide answer

    A: A rotation of \(180^\circ\) about the centre

    Every point goes to the opposite side of the centre at the same distance.

  9. 9

    The point \((1, 3)\) is enlarged by scale factor \(-2\) with centre \((0, 0)\). What is the image?

    1. A\((2, 6)\)
    2. B\((-1, -3)\)
    3. C\((-2, 6)\)
    4. D\((-2, -6)\)
    Show answerHide answer

    D: \((-2, -6)\)

    Multiply both coordinates by \(-2\).

Combined Transformations

Just this lesson
  1. 1 Describe [4 marks]

    Triangle \(A\) is reflected in the \(y\)-axis to give triangle \(B\). Triangle \(B\) is reflected in the \(x\)-axis to give triangle \(C\). (a) Draw triangles \(B\) and \(C\). (2 marks) (b) Describe fully the single transformation that maps \(A\) onto \(C\). (2 marks)

    Triangle A on a grid in the first quadrant.
    Show answerHide answer

    Model answer

    (a) \(B\) has vertices \((-1, 2)\), \((-3, 2)\) and \((-1, 5)\), and \(C\) has vertices \((-1, -2)\), \((-3, -2)\) and \((-1, -5)\). (b) \((x, y)\) goes to \((-x, -y)\), which is a rotation of \(180^\circ\) about the origin.

    Mark scheme

    • (a) \(B\) correct — B1
    • (a) \(C\) correct — B1
    • (b) Rotation, \(180^\circ\) — B1
    • (b) About the origin — B1
  2. 2 Describe [3 marks]

    The point \((2, 3)\) is reflected in the line \(x = 1\), and the image is then reflected in the line \(x = 5\). Describe the single transformation that has the same effect.

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    Model answer

    The first reflection gives \((0, 3)\) and the second gives \((10, 3)\). The point has moved 8 to the right, which is twice the distance between the lines, \(2 \times 4 = 8\). So it is a translation by \(\begin{pmatrix} 8 \\ 0 \end{pmatrix}\).

    Mark scheme

    • \((0, 3)\) seen — M1
    • \((10, 3)\) seen — M1
    • Translation by \(\begin{pmatrix} 8 \\ 0 \end{pmatrix}\) — A1
  3. 3 Write down [2 marks]

    (a) A shape is rotated through \(90^\circ\) about the point \((2, 1)\). Write down the coordinates of the point that does not move. (1 mark) (b) A shape is reflected in the line \(y = 3\). Describe the points that do not move. (1 mark)

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    Model answer

    (a) The centre of rotation, \((2, 1)\), is invariant. (b) Every point on the line \(y = 3\) stays where it is.

    Mark scheme

    • (a) \((2, 1)\) — B1
    • (b) The points on the line \(y = 3\) — B1
  4. 4 Work out [3 marks]

    The point \(P\) is \((3, 1)\). \(P\) is translated by the vector \(\begin{pmatrix} -2 \\ 3 \end{pmatrix}\) and the image is then reflected in the \(x\)-axis. Write down the coordinates of the final image.

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    Model answer

    The translation gives \((1, 4)\), and the reflection in the \(x\)-axis gives \((1, -4)\).

    Mark scheme

    • \((1, 4)\) — B1
    • Changes the sign of the \(y\)-coordinate of their image — M1
    • \((1, -4)\) — A1
  5. 5 Show that [3 marks]

    Show that a reflection in the line \(y = x\) followed by a reflection in the line \(y = -x\) is the same as a rotation of \(180^\circ\) about the origin. Use the point \((2, 5)\).

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    Model answer

    Reflecting \((2, 5)\) in \(y = x\) swaps the coordinates, giving \((5, 2)\). Reflecting in \(y = -x\) swaps and changes both signs, giving \((-2, -5)\). A \(180^\circ\) rotation about the origin sends \((2, 5)\) to \((-2, -5)\), the same point.

    Mark scheme

    • \((5, 2)\) seen — M1
    • \((-2, -5)\) seen — M1
    • Compares with the \(180^\circ\) rotation, \((-2, -5)\) — C1
  6. 6 Describe [4 marks]

    The point \((4, -1)\) is rotated through \(90^\circ\) clockwise about the origin, and the image is then reflected in the \(y\)-axis. Describe fully the single transformation that has the same effect.

    Show answerHide answer

    Model answer

    The rotation gives \((-1, -4)\), and the reflection gives \((1, -4)\). In general \((x, y)\) goes to \((y, -x)\) and then to \((-y, -x)\). That is a reflection in the line \(y = -x\).

    Mark scheme

    • \((-1, -4)\) seen — M1
    • \((1, -4)\) seen — M1
    • Reflection — A1
    • In the line \(y = -x\) — A1

Quick check

  1. 1

    What single transformation is a reflection in the \(x\)-axis followed by a reflection in the \(y\)-axis?

    1. AA rotation of \(180^\circ\) about the origin
    2. BA translation
    3. CA reflection in the line \(y = x\)
    4. DAn enlargement with scale factor \(-1\)
    Show answerHide answer

    A: A rotation of \(180^\circ\) about the origin

    \((x, y)\) goes to \((x, -y)\) and then to \((-x, -y)\), which is a half turn.

  2. 2

    What do two reflections in parallel lines give?

    1. AA rotation
    2. BA reflection
    3. CAn enlargement
    4. DA translation
    Show answerHide answer

    D: A translation

    The shape is moved in a straight line, at right angles to the mirror lines.

  3. 3

    The point \((1, 1)\) is reflected in \(x = 2\) and then in \(x = 5\). Which translation has the same effect?

    1. A\(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\)
    2. B\(\begin{pmatrix} 4 \\ 0 \end{pmatrix}\)
    3. C\(\begin{pmatrix} 6 \\ 0 \end{pmatrix}\)
    4. D\(\begin{pmatrix} 0 \\ 6 \end{pmatrix}\)
    Show answerHide answer

    C: \(\begin{pmatrix} 6 \\ 0 \end{pmatrix}\)

    The lines are 3 apart, and the translation is twice that distance, 6, at right angles to them.

  4. 4

    Which points are invariant in a reflection?

    1. AEvery point
    2. BThe points on the mirror line
    3. COnly the origin
    4. DThere are none
    Show answerHide answer

    B: The points on the mirror line

    Points on the mirror line do not move.

  5. 5

    Which point is invariant in a rotation?

    1. AThe centre of rotation
    2. BEvery point on the shape
    3. CThe corner of the shape
    4. DThere are none
    Show answerHide answer

    A: The centre of rotation

    The centre stays fixed while everything else turns around it.

  6. 6

    The point \((2, 1)\) is reflected in the line \(y = x\) and then in the \(x\)-axis. What is the final image?

    1. A\((-1, 2)\)
    2. B\((2, -1)\)
    3. C\((-2, -1)\)
    4. D\((1, -2)\)
    Show answerHide answer

    D: \((1, -2)\)

    Swapping gives \((1, 2)\), and changing the sign of \(y\) gives \((1, -2)\).

  7. 7

    Two mirror lines cross at a point, with an angle of \(30^\circ\) between them. What single transformation is a reflection in one followed by the other?

    1. AA rotation of \(30^\circ\) about the crossing point
    2. BA translation of 30 units
    3. CA rotation of \(60^\circ\) about the crossing point
    4. DA reflection in the crossing point
    Show answerHide answer

    C: A rotation of \(60^\circ\) about the crossing point

    The rotation is through twice the angle between the lines.

  8. 8

    Does a translation have an invariant point?

    1. AYes, the origin
    2. BNo, every point moves
    3. CYes, the centre
    4. DOnly for small shapes
    Show answerHide answer

    B: No, every point moves

    A translation moves every point by the same vector.

  9. 9

    Two rotations of \(90^\circ\) clockwise about the same centre are carried out one after the other. What single transformation is this?

    1. AA rotation of \(180^\circ\) about that centre
    2. BA rotation of \(90^\circ\) about that centre
    3. CA reflection
    4. DA translation
    Show answerHide answer

    A: A rotation of \(180^\circ\) about that centre

    \(90^\circ + 90^\circ = 180^\circ\).

Congruence and Similarity

Just this lesson
  1. 1 Give a reason [2 marks]

    Triangle \(ABC\) has sides of 3 cm, 4 cm and 5 cm. Triangle \(PQR\) has sides of 3 cm, 4 cm and 5 cm. Are the triangles congruent? Give a reason for your answer.

    Show answerHide answer

    Model answer

    Yes. All three sides of one triangle are equal to the three sides of the other, so they are congruent by SSS.

    Mark scheme

    • Yes — B1
    • All three pairs of sides are equal, or SSS — C1
  2. 2 Prove [4 marks]

    \(ABCD\) is a kite with \(AB = AD\) and \(BC = DC\). (a) Prove that triangles \(ABC\) and \(ADC\) are congruent. (3 marks) (b) Hence write down the size of angle \(ABC\) compared with angle \(ADC\). (1 mark)

    A kite ABCD with AB equal to AD, BC equal to DC and the diagonal AC drawn.
    Show answerHide answer

    Model answer

    (a) \(AB = AD\) (given), \(BC = DC\) (given) and \(AC\) is a side of both triangles. All three sides are equal, so the triangles are congruent by SSS. (b) Corresponding angles in congruent triangles are equal, so angle \(ABC\) equals angle \(ADC\).

    Mark scheme

    • \(AB = AD\) and \(BC = DC\) given — M1
    • \(AC\) is a common side — M1
    • SSS, so the triangles are congruent — C1
    • (b) Angle \(ABC\) = angle \(ADC\) — B1
  3. 3 Work out [3 marks]

    Triangles \(ABC\) and \(DEF\) are similar. (a) Work out the scale factor of the enlargement from \(ABC\) to \(DEF\). (1 mark) (b) Work out the length of \(EF\). (1 mark) (c) Work out the length of \(FD\). (1 mark)

    Two similar triangles ABC and DEF, with the sides of ABC 5, 6 and 7 centimetres and DE equal to 10 centimetres.
    Show answerHide answer

    Model answer

    (a) \(DE\) matches \(AB\), so the scale factor is \(\dfrac{10}{5} = 2\). (b) \(EF = 6 \times 2 = 12\) cm. (c) \(FD = 7 \times 2 = 14\) cm.

    Mark scheme

    • (a) 2 — B1
    • (b) 12 — B1
    • (c) 14 — B1
  4. 4 Work out [2 marks]

    Triangles \(ABC\) and \(DEF\) are similar. \(AB = 4\) cm, \(BC = 6\) cm and \(DE = 10\) cm, where \(DE\) corresponds to \(AB\) and \(EF\) corresponds to \(BC\). Work out the length of \(EF\).

    Show answerHide answer

    Model answer

    The scale factor is \(\dfrac{10}{4} = 2.5\), so \(EF = 6 \times 2.5 = 15\) cm.

    Mark scheme

    • \(\dfrac{10}{4}\) or 2.5 — M1
    • 15 cm — A1
  5. 5 Work out [3 marks]

    Two similar shapes have lengths in the ratio \(2 : 3\). The area of the smaller shape is 20 cm\(^2\). Work out the area of the larger shape.

    Show answerHide answer

    Model answer

    The area scale factor is \(\left(\dfrac{3}{2}\right)^2 = \dfrac{9}{4}\). The larger area is \(20 \times \dfrac{9}{4} = 45\) cm\(^2\).

    Mark scheme

    • \(\left(\dfrac{3}{2}\right)^2\) or \(\dfrac{9}{4}\) — M1
    • \(20 \times \dfrac{9}{4}\) — M1
    • 45 cm\(^2\) — A1
  6. 6 Work out [3 marks]

    Two similar solids have volumes in the ratio \(8 : 27\). The surface area of the smaller solid is 36 cm\(^2\). Work out the surface area of the larger solid.

    Show answerHide answer

    Model answer

    The volumes are in the ratio \(8 : 27\), so the lengths are in the ratio \(\sqrt[3]{8} : \sqrt[3]{27} = 2 : 3\). The area ratio is \(4 : 9\), so the larger surface area is \(36 \times \dfrac{9}{4} = 81\) cm\(^2\).

    Mark scheme

    • Length ratio \(2 : 3\) — M1
    • Area ratio \(4 : 9\) or the scale factor \(\dfrac{9}{4}\) — M1
    • 81 cm\(^2\) — A1

Quick check

  1. 1

    Which of these is not enough to show that two triangles are congruent?

    1. AThree equal sides
    2. BThree equal angles
    3. CTwo sides and the angle between them
    4. DA right angle, the hypotenuse and another side
    Show answerHide answer

    B: Three equal angles

    Three equal angles give similar triangles, which may be different sizes.

  2. 2

    Two triangles have two equal sides and the equal angle between them. Which condition is this?

    1. ASAS
    2. BSSS
    3. CASA
    4. DRHS
    Show answerHide answer

    A: SAS

    Side, angle, side with the angle between the sides is SAS.

  3. 3

    Two similar triangles have matching sides of 6 cm and 15 cm. What is the scale factor from the smaller to the larger?

    1. A\(9\)
    2. B\(0.4\)
    3. C\(90\)
    4. D\(2.5\)
    Show answerHide answer

    D: \(2.5\)

    \(\dfrac{15}{6} = 2.5\).

  4. 4

    A triangle with sides 3, 4 and 5 cm is similar to a triangle with sides 9 cm, \(x\) cm and 15 cm. What is \(x\)?

    1. A\(8\)
    2. B\(10\)
    3. C\(12\)
    4. D\(7\)
    Show answerHide answer

    C: \(12\)

    The scale factor is \(\dfrac{9}{3} = 3\), so \(x = 4 \times 3 = 12\).

  5. 5

    Triangles \(ABC\) and \(DEF\) are similar, with \(AB = 4\), \(DE = 10\) and \(BC = 6\). What is \(EF\)?

    1. A4 cm
    2. B15 cm
    3. C9.6 cm
    4. D7.5 cm
    Show answerHide answer

    B: 15 cm

    The scale factor is \(\dfrac{10}{4} = 2.5\), so \(EF = 6 \times 2.5 = 15\) cm.

  6. 6

    Two triangles have all three angles equal. What can you say?

    1. AThey are similar
    2. BThey are congruent
    3. CThey have the same area
    4. DThey are enlargements with scale factor 1
    Show answerHide answer

    A: They are similar

    Equal angles make the shapes similar, but not necessarily the same size.

  7. 7

    The lengths of a shape are doubled. By what factor is the area multiplied?

    1. A2
    2. B8
    3. C16
    4. D4
    Show answerHide answer

    D: 4

    The area scale factor is \(2^2 = 4\).

  8. 8

    The lengths of a solid are multiplied by 3. By what factor is the volume multiplied?

    1. A9
    2. B3
    3. C27
    4. D6
    Show answerHide answer

    C: 27

    The volume scale factor is \(3^3 = 27\).

  9. 9

    Two similar shapes have areas in the ratio \(9 : 25\). What is the ratio of their lengths?

    1. A\(9 : 25\)
    2. B\(3 : 5\)
    3. C\(81 : 625\)
    4. D\(4.5 : 12.5\)
    Show answerHide answer

    B: \(3 : 5\)

    Take square roots: \(\sqrt{9} : \sqrt{25} = 3 : 5\).