Rotations

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)

   

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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 \(B\).

   

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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\((-3, -2)\)
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   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\((5, 2)\)
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   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\((1, -3)\)
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   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\((-4, 2)\)
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   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\((-2, -1)\)
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   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\((6, 6)\)
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   C: \((3, 3)\)

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