Transformations: Reflection and Rotation
A reflection creates a mirror image of a shape in a mirror line, with every point of the image the same perpendicular distance from the line as the matching point on the object, on the opposite side, while a rotation turns a shape through a given angle, in a given direction, about a fixed point called the centre of rotation. Both transformations produce an image congruent to the original object: identical in size and shape, just in a different position or orientation.
Before you start
Make sure you're comfortable with these topics first:
Method
- For a reflection, identify the mirror line, for example the x-axis, the y-axis, or a line such as y = x.
- Reflect each vertex of the shape in turn: measure its perpendicular distance to the mirror line, then plot the image point the same distance on the other side.
- For reflection in the x-axis, the x-coordinate stays the same and the y-coordinate changes sign; for reflection in the y-axis, the y-coordinate stays the same and the x-coordinate changes sign.
- For a rotation, identify the centre of rotation, the angle of turn (usually 90, 180 or 270 degrees), and the direction (clockwise or anticlockwise).
- Rotate each vertex about the centre, using tracing paper or a coordinate rule for that angle, then join the image points in the same order as the object.
- Always describe a reflection by its mirror line, and a rotation by its centre, angle and direction; a description missing any of these parts is incomplete.
Worked example
Triangle T has vertices A(1, 1), B(4, 1) and C(1, 3). (a) Reflect triangle T in the y-axis to give triangle T1. (b) State the coordinates of the image of A, B and C after this reflection.
- Reflecting in the y-axis keeps the y-coordinate the same and changes the sign of the x-coordinate: (x, y) becomes (-x, y).
- A(1, 1) maps to (-1, 1).
- B(4, 1) maps to (-4, 1).
- C(1, 3) maps to (-1, 3), so triangle T1 has vertices (-1, 1), (-4, 1) and (-1, 3).
Practice questions
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Q1Reflect the point (5, 2) in the x-axis. State the image coordinates.Show answer
Answer: (5, -2)
Q2Reflect the point (-3, 6) in the y-axis. State the image coordinates.Show answer
Answer: (3, 6)
Q3Reflect the point (4, 7) in the line y = x. State the image coordinates.Show answer
Answer: (7, 4), since reflecting in y = x swaps the x- and y-coordinates
Q4Point D(2, 5) is rotated 90 degrees clockwise about the origin. State the image coordinates.Show answer
Answer: (5, -2)
Q5Point E(3, -4) is rotated 180 degrees about the origin. State the image coordinates.Show answer
Answer: (-3, 4)
Q6Point F(-2, 6) is rotated 90 degrees anticlockwise about the origin. State the image coordinates.Show answer
Answer: (-6, -2)
Q7Point (6, 1) is reflected in a horizontal mirror line to give the image point (6, -1). State the equation of the mirror line.Show answer
Answer: The mirror line lies halfway between y = 1 and y = -1, so its equation is y = 0 (the x-axis)
Q8A rotation about the origin maps the point (2, 3) to the point (-3, 2). State the angle and direction of the rotation.Show answer
Answer: 90 degrees anticlockwise (using (x, y) maps to (-y, x) for a 90 degree anticlockwise rotation about the origin)
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
Triangle P has vertices (2, 1), (6, 1) and (2, 5). Triangle P is reflected in the line x = 1 to give triangle Q. (a) State the coordinates of the image of each vertex. (b) State one property that is the same for triangle P and triangle Q, and one property that is different.
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Shape S is rotated 180 degrees about the point (2, 2) to give shape S1. One vertex of shape S is at (5, 6). (a) Find the coordinates of the image of this vertex. (b) State the mathematical name for the single transformation that would map S1 back to S.
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Free printable worksheet
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This topic is chapter 16 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
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