GCSE Maths: Rotation (Higher)

Key words. A rotation turns a shape about a fixed point called the centre of rotation. We state the angle and the direction (clockwise or anticlockwise).

Method. To rotate a shape about C, write each vertex as a move from C, turn that move through the angle, and add it back on to C.

1. Mark the centre of rotation C on the grid. Every point of the shape turns around C, so we always measure from C. The centre itself never moves.
   C = (3, 2)
2. Write each vertex as a move from C. A turn is easiest to do by thinking about the move (across, up) needed to get from C to the point.
   (1, 1): (−2, −1) means 2 left, 1 down
   (4, 2): (1, 0) means 1 right
   (2, 4): (−1, 2) means 1 left, 2 up
3. Turn each move by 180°. A half turn sends every point to the opposite side of C, the same distance away, so both numbers change sign: (a, b) → (−a, −b).
   (−2, −1) → (2, 1)
   (1, 0) → (−1, 0)
   (−1, 2) → (1, −2)
4. Add the turned move to C to find each image vertex. The turned move is measured from C, so we start at C again.
   C + (2, 1) = (5, 3)
   C + (−1, 0) = (2, 2)
   C + (1, −2) = (4, 0)
5. Plot the image points and join them up to make shape B. The image is the same size and shape as the object but turned.
   B has vertices at (5, 3), (2, 2), (4, 0).

Check. In a half turn the centre is the midpoint of a point and its image. The midpoint of (1, 1) and (5, 3) is (3, 2), which is C. ✓

Watch out. Turn about the centre C, not about the origin (unless C is the origin). And check the direction: clockwise is the way the hands of a clock move.

Key words. A rotation turns a shape about a fixed point called the centre of rotation. We state the angle and the direction (clockwise or anticlockwise).

Method. To rotate a shape about C, write each vertex as a move from C, turn that move through the angle, and add it back on to C.

1. Mark the centre of rotation C on the grid. Every point of the shape turns around C, so we always measure from C. The centre itself never moves.
   C = (1, 2)
2. Write each vertex as a move from C. A turn is easiest to do by thinking about the move (across, up) needed to get from C to the point.
   (2, 1): (1, −1) means 1 right, 1 down
   (4, 1): (3, −1) means 3 right, 1 down
   (4, 2): (3, 0) means 3 right
   (3, 2): (2, 0) means 2 right
   (3, 4): (2, 2) means 2 right, 2 up
   (2, 4): (1, 2) means 1 right, 2 up
3. Turn each move by 90° anticlockwise. A quarter turn anticlockwise changes a move using the rule (a, b) → (−b, a). Distances from C stay the same; only the direction changes.
   (1, −1) → (1, 1)
   (3, −1) → (1, 3)
   (3, 0) → (0, 3)
   (2, 0) → (0, 2)
   (2, 2) → (−2, 2)
   (1, 2) → (−2, 1)
4. Add the turned move to C to find each image vertex. The turned move is measured from C, so we start at C again.
   C + (1, 1) = (2, 3)
   C + (1, 3) = (2, 5)
   C + (0, 3) = (1, 5)
   C + (0, 2) = (1, 4)
   C + (−2, 2) = (−1, 4)
   C + (−2, 1) = (−1, 3)
5. Plot the image points and join them up to make shape B. The image is the same size and shape as the object but turned.
   B has vertices at (2, 3), (2, 5), (1, 5), (1, 4), (−1, 4), (−1, 3).

Check. After a quarter turn the line from C to (4, 1) and the line from C to its image (2, 5) are at right angles, so the "dot product" of the two moves is 0: (3 × 1) + (−1 × 3) = 0. Both are also the same distance from C. ✓

Watch out. Turn about the centre C, not about the origin (unless C is the origin). And check the direction: clockwise is the way the hands of a clock move.

Key words. A rotation turns a shape about a fixed point called the centre of rotation. We state the angle and the direction (clockwise or anticlockwise).

Method. To rotate a shape about C, write each vertex as a move from C, turn that move through the angle, and add it back on to C.

1. Mark the centre of rotation C on the grid. Every point of the shape turns around C, so we always measure from C. The centre itself never moves.
   C = (4, 0)
2. Write each vertex as a move from C. A turn is easiest to do by thinking about the move (across, up) needed to get from C to the point.
   (1, 1): (−3, 1) means 3 left, 1 up
   (4, 1): (0, 1) means 1 up
   (3, 3): (−1, 3) means 1 left, 3 up
   (1, 3): (−3, 3) means 3 left, 3 up
3. Turn each move by 90° clockwise. A quarter turn clockwise changes a move using the rule (a, b) → (b, −a). Distances from C stay the same; only the direction changes.
   (−3, 1) → (1, 3)
   (0, 1) → (1, 0)
   (−1, 3) → (3, 1)
   (−3, 3) → (3, 3)
4. Add the turned move to C to find each image vertex. The turned move is measured from C, so we start at C again.
   C + (1, 3) = (5, 3)
   C + (1, 0) = (5, 0)
   C + (3, 1) = (7, 1)
   C + (3, 3) = (7, 3)
5. Plot the image points and join them up to make shape B. The image is the same size and shape as the object but turned.
   B has vertices at (5, 3), (5, 0), (7, 1), (7, 3).

Check. After a quarter turn the line from C to (1, 3) and the line from C to its image (7, 3) are at right angles, so the "dot product" of the two moves is 0: (−3 × 3) + (3 × 3) = 0. Both are also the same distance from C. ✓

Watch out. Turn about the centre C, not about the origin (unless C is the origin). And check the direction: clockwise is the way the hands of a clock move.

Key words. A transformation changes the position or size of a shape. The original shape is the object; the new shape is the image.

Method. Apply the transformation to the coordinates of P.

1. Remember what this transformation does. Rotate about (1, 1): write each point as a move from the centre, turn the move using (a, b) → (−b, a), then add the centre back on.
   Rotation of 90° anticlockwise about (1, 1)
2. Apply it to the point. Follow the rule using the coordinates of the point.
   (2, 3): move from centre (1, 2) → turned (−2, 1) → (−1, 2)
3. Write the coordinates of the image. The image of P is called P'.
   P' = (−1, 2)

Check. Look at the picture: P and P' are placed as the transformation requires. ✓

Watch out. Take care with minus signs: write the new coordinates carefully, one at a time.

Key words. A rotation turns a shape about a fixed point called the centre of rotation. We state the angle and the direction (clockwise or anticlockwise).

Method. To rotate a shape about C, write each vertex as a move from C, turn that move through the angle, and add it back on to C.

1. Mark the centre of rotation C on the grid. Every point of the shape turns around C, so we always measure from C. The centre itself never moves.
   C = (1, 1)
2. Write each vertex as a move from C. A turn is easiest to do by thinking about the move (across, up) needed to get from C to the point.
   (2, 2): (1, 1) means 1 right, 1 up
   (4, 2): (3, 1) means 3 right, 1 up
   (5, 4): (4, 3) means 4 right, 3 up
   (2, 3): (1, 2) means 1 right, 2 up
3. Turn each move by 90° clockwise. A quarter turn clockwise changes a move using the rule (a, b) → (b, −a). Distances from C stay the same; only the direction changes.
   (1, 1) → (1, −1)
   (3, 1) → (1, −3)
   (4, 3) → (3, −4)
   (1, 2) → (2, −1)
4. Add the turned move to C to find each image vertex. The turned move is measured from C, so we start at C again.
   C + (1, −1) = (2, 0)
   C + (1, −3) = (2, −2)
   C + (3, −4) = (4, −3)
   C + (2, −1) = (3, 0)
5. Plot the image points and join them up to make shape B. The image is the same size and shape as the object but turned.
   B has vertices at (2, 0), (2, −2), (4, −3), (3, 0).

Check. After a quarter turn the line from C to (5, 4) and the line from C to its image (4, −3) are at right angles, so the "dot product" of the two moves is 0: (4 × 3) + (3 × −4) = 0. Both are also the same distance from C. ✓

Watch out. Turn about the centre C, not about the origin (unless C is the origin). And check the direction: clockwise is the way the hands of a clock move.

Key words. A rotation turns a shape about a fixed point called the centre of rotation. We state the angle and the direction (clockwise or anticlockwise).

Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).

1. Decide which type of transformation it is. A rotation turns the shape. Compare with a translation, where all sides stay parallel to the originals.
   Compare a side: it is 3 long in A and 3 long in B.
   The size is unchanged, B is not flipped, but its sides are not parallel to the originals: the shape has turned. So the transformation is a rotation.
2. Find the centre of rotation. The centre is the same distance from a point and from its image. The origin is the first place to try. It works if the squared distances from the origin to the point and its image match.
   Distance² of (1, 1) from (0, 0) = 1² + 1² = 2
   Distance² of (1, −1) from (0, 0) = 1² + −1² = 2
   They match, and the same works for the other vertices, so the centre is (0, 0).
3. Find the angle and direction. Write the move from C to a vertex, and the move from C to its image. A quarter turn changes (a, b) in a special way: clockwise gives (b, −a) and anticlockwise gives (−b, a).
   From C to (1, 1): (1, 1) (1 right, 1 up)
   From C to (1, −1): (1, −1) (1 right, 1 down)
   This matches (a, b) → (b, −a), so the turn is 90° clockwise.
4. Write the full description. We have found every piece of information needed for this transformation.
   Rotation, 90° clockwise, centre (0, 0)

Check. Rotate (4, 1) about (0, 0) by 90° clockwise: we get (1, −4), which matches the vertex (1, −4) of B. ✓

Watch out. Give three things for a rotation: the angle, the direction (clockwise or anticlockwise) and the centre. A 180° turn does not need a direction.

Key words. A rotation turns a shape about a fixed point called the centre of rotation. We state the angle and the direction (clockwise or anticlockwise).

Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).

1. Decide which type of transformation it is. A rotation turns the shape. Compare with a translation, where all sides stay parallel to the originals.
   Compare the areas: 4 square units for A and 4 for B.
   The size is unchanged, B is not flipped, but its sides are not parallel to the originals: the shape has turned. So the transformation is a rotation.
2. Find the centre of rotation. In a half turn, the centre is exactly half-way between a point and its image. Find the midpoint for two different vertices: they should be the same point.
   (2, 2) and (−2, −2): midpoint (0, 0)
   (5, 3) and (−5, −3): midpoint (0, 0)
   Both midpoints are the same, so the centre is (0, 0).
3. Find the angle and direction. From C the moves to the vertex and to its image point in exactly opposite directions, so it is a half turn. For a half turn the direction does not matter.
   From C to (2, 2): (2, 2); from C to (−2, −2): (−2, −2). Both signs have changed, so the angle is 180°.
4. Write the full description. We have found every piece of information needed for this transformation.
   Rotation, 180°, centre (0, 0)

Check. Rotate (5, 3) about (0, 0) by 180°: we get (−5, −3), which matches the vertex (−5, −3) of B. ✓

Watch out. Give three things for a rotation: the angle, the direction (clockwise or anticlockwise) and the centre. A 180° turn does not need a direction.

Key words. A rotation turns a shape about a fixed point called the centre of rotation. We state the angle and the direction (clockwise or anticlockwise).

Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).

1. Decide which type of transformation it is. A rotation turns the shape. Compare with a translation, where all sides stay parallel to the originals.
   Compare a side: it is 3 long in A and 3 long in B.
   The size is unchanged, B is not flipped, but its sides are not parallel to the originals: the shape has turned. So the transformation is a rotation.
2. Find the centre of rotation. The centre of rotation is the only point that does not move. A vertex that is in the same place on the object and the image must be the centre.
   The vertex (1, 1) is in the same place on A and B, so the centre is (1, 1).
3. Find the angle and direction. Write the move from C to a vertex, and the move from C to its image. A quarter turn changes (a, b) in a special way: clockwise gives (b, −a) and anticlockwise gives (−b, a).
   From C to (1, 1): (0, 0) (no move)
   From C to (1, 1): (0, 0) (no move)
   This matches (a, b) → (−b, a), so the turn is 90° anticlockwise.
4. Write the full description. We have found every piece of information needed for this transformation.
   Rotation, 90° anticlockwise, centre (1, 1)

Check. Rotate (4, 1) about (1, 1) by 90° anticlockwise: we get (1, 4), which matches the vertex (1, 4) of B. ✓

Watch out. Give three things for a rotation: the angle, the direction (clockwise or anticlockwise) and the centre. A 180° turn does not need a direction.

Key words. A rotation turns a shape about a fixed point called the centre of rotation. We state the angle and the direction (clockwise or anticlockwise).

Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).

1. Decide which type of transformation it is. A rotation turns the shape. Compare with a translation, where all sides stay parallel to the originals.
   Compare a side: it is 3 long in A and 3 long in B.
   The size is unchanged, B is not flipped, but its sides are not parallel to the originals: the shape has turned. So the transformation is a rotation.
2. Find the centre of rotation. In a half turn, the centre is exactly half-way between a point and its image. Find the midpoint for two different vertices: they should be the same point.
   (2, 1) and (4, 3): midpoint (3, 2)
   (4, 1) and (2, 3): midpoint (3, 2)
   Both midpoints are the same, so the centre is (3, 2).
3. Find the angle and direction. From C the moves to the vertex and to its image point in exactly opposite directions, so it is a half turn. For a half turn the direction does not matter.
   From C to (2, 1): (−1, −1); from C to (4, 3): (1, 1). Both signs have changed, so the angle is 180°.
4. Write the full description. We have found every piece of information needed for this transformation.
   Rotation, 180°, centre (3, 2)

Check. Rotate (4, 1) about (3, 2) by 180°: we get (2, 3), which matches the vertex (2, 3) of B. ✓

Watch out. Give three things for a rotation: the angle, the direction (clockwise or anticlockwise) and the centre. A 180° turn does not need a direction.

Method. Use the rule for a clockwise quarter turn about the origin: (a, b) → (b, −a).

1. Apply the rule. For (3, 1) we have a = 3 and b = 1.
   (3, 1) → (1, −3)
2. Compare with Ben. Ben’s answer (−1, 3) uses the rule (a, b) → (−b, a), which is the rule for 90° anticlockwise.
   Ben turned the wrong way.
3. Think about it with a picture. (3, 1) is to the right of the origin. Turning clockwise, the point swings down, to the bottom right. (1, −3) is in the bottom right; (−1, 3) is in the top left.
   Correct image: (1, −3)

Check. Distance check: 3² + 1² = 10 and 1² + (−3)² = 10, so both points are the same distance from O. ✓

Watch out. Clockwise means the way a clock’s hands go. Sketch the point and swing it round to check.

Method. Compare the coordinates of P and P′ with the three turning rules.

1. Write the point and its image. The centre is the origin, so the coordinates are the moves from the centre.
   (2, 5) → (5, −2)
2. Match the rule. (a, b) → (b, −a) is the rule for 90° clockwise. Here a = 2 and b = 5, so (b, −a) = (5, −2).
   This matches.
3. Write the description. Give angle, direction and centre.
   Rotation, 90° clockwise, centre (0, 0)

Check. Rotating (2, 5) 90° anticlockwise would give (−5, 2), which is not P′. ✓

Watch out. A full description needs all three: angle, direction, centre.

Method. Do the two quarter turns one after the other and compare the answer with a half turn.

1. First 90° clockwise. (a, b) → (b, −a).
   (3, 1) → (1, −3)
2. Second 90° clockwise. Use the same rule on the new point.
   (1, −3) → (−3, −1)
3. Half turn. (a, b) → (−a, −b).
   (3, 1) → (−3, −1)
4. Compare. Both routes end at (−3, −1), so two quarter turns make a half turn.
   The results are the same.

Check. Quarter turn + quarter turn = 90° + 90° = 180°. ✓

Watch out. One example shows that it works for this point. For a proof, use (a, b) → (b, −a) → (−a, −b).

Key words. A rotation turns a shape about a fixed point called the centre of rotation. We state the angle and the direction (clockwise or anticlockwise).

Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).

1. Decide which type of transformation it is. A rotation turns the shape. Compare with a translation, where all sides stay parallel to the originals.
   Compare a side: it is 3 long in A and 3 long in B.
   The size is unchanged, B is not flipped, but its sides are not parallel to the originals: the shape has turned. So the transformation is a rotation.
2. Find the centre of rotation using perpendicular bisectors. The centre is the same distance from a point and from its image. So it lies on the perpendicular bisector of the line joining them (the line at right angles through the midpoint). The centre is where two such bisectors cross.
   For (−3, 1) → (0, 6): the points (x, y) that are equally far from both satisfy 3x + 5y = 13.
   For (0, 1) → (0, 3): the points equally far from both satisfy y = 2.
   Solving the two equations together gives x = 1, y = 2.
   Check in both: 3(1) + 5(2) = 13 ✓; 0(1) + 1(2) = 2 ✓.
   The centre is (1, 2).
3. Find the angle and direction. Write the move from C to a vertex, and the move from C to its image. A quarter turn changes (a, b) in a special way: clockwise gives (b, −a) and anticlockwise gives (−b, a).
   From C to (−3, 1): (−4, −1) (4 left, 1 down)
   From C to (0, 6): (−1, 4) (1 left, 4 up)
   This matches (a, b) → (b, −a), so the turn is 90° clockwise.
4. Write the full description. We have found every piece of information needed for this transformation.
   Rotation, 90° clockwise, centre (1, 2)

Check. Rotate (0, 1) about (1, 2) by 90° clockwise: we get (0, 3), which matches the vertex (0, 3) of B. ✓

Watch out. Give three things for a rotation: the angle, the direction (clockwise or anticlockwise) and the centre. A 180° turn does not need a direction.

Key words. A rotation turns a shape about a fixed point called the centre of rotation. We state the angle and the direction (clockwise or anticlockwise).

Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).

1. Decide which type of transformation it is. A rotation turns the shape. Compare with a translation, where all sides stay parallel to the originals.
   Compare the areas: 4 square units for A and 4 for B.
   The size is unchanged, B is not flipped, but its sides are not parallel to the originals: the shape has turned. So the transformation is a rotation.
2. Find the centre of rotation using perpendicular bisectors. The centre is the same distance from a point and from its image. So it lies on the perpendicular bisector of the line joining them (the line at right angles through the midpoint). The centre is where two such bisectors cross.
   For (1, 2) → (−1, 4): the points (x, y) that are equally far from both satisfy x − y = −3.
   For (4, 3) → (−2, 7): the points equally far from both satisfy 3x − 2y = −7.
   Solving the two equations together gives x = −1, y = 2.
   Check in both: 1(−1) + −1(2) = −3 ✓; 3(−1) + −2(2) = −7 ✓.
   The centre is (−1, 2).
3. Find the angle and direction. Write the move from C to a vertex, and the move from C to its image. A quarter turn changes (a, b) in a special way: clockwise gives (b, −a) and anticlockwise gives (−b, a).
   From C to (1, 2): (2, 0) (2 right)
   From C to (−1, 4): (0, 2) (2 up)
   This matches (a, b) → (−b, a), so the turn is 90° anticlockwise.
4. Write the full description. We have found every piece of information needed for this transformation.
   Rotation, 90° anticlockwise, centre (−1, 2)

Check. Rotate (4, 3) about (−1, 2) by 90° anticlockwise: we get (−2, 7), which matches the vertex (−2, 7) of B. ✓

Watch out. Give three things for a rotation: the angle, the direction (clockwise or anticlockwise) and the centre. A 180° turn does not need a direction.

Method. Let C = (c, d). A clockwise quarter turn about C changes the move (a, b) from C into (b, −a). Write this for P and P′ to get two equations.

1. Write the move from C to P and the move from C to P′. Move = point − centre.
   C to P: (3 − c, 4 − d)
   C to P′: (6 − c, 1 − d)
2. Apply the clockwise rule to the first move. (a, b) → (b, −a), so the turned move is (4 − d, −(3 − c)).
   Turned move: (4 − d, c − 3)
3. Match it with the move to P′. The turned move must equal C to P′.
   6 − c = 4 − d  →  d − c = −2
   1 − d = c − 3  →  c + d = 4
4. Solve the two equations. Add them: (d − c) + (c + d) = −2 + 4, so 2d = 2.
   d = 1, then c = 4 − 1 = 3
   C = (3, 1)

Check. From C(3, 1): the move to P is (0, 3). Turning 90° clockwise gives (3, 0), and (3, 1) + (3, 0) = (6, 1) = P′. ✓

Watch out. Write moves as “point minus centre”, never “centre minus point”, or the signs go wrong.