A reflection flips a shape over a mirror line; a rotation turns it about a fixed centre. Both keep the shape exactly the same size, so the image is congruent to the object.
To describe a reflection fully, give the equation of the mirror line. To describe a rotation fully, give the angle, the direction (clockwise or anticlockwise) and the centre. Always describe ONE single transformation.
Fig. 1: triangle A is reflected in the y-axis to give B, and rotated 90 degrees clockwise about the origin to give C; all three are congruent
Section 01
Mirror, mirror.
In the lab, drag the mirror line to a vertical position such as \(x = 1\), then to a horizontal one such as \(y = -1\). Watch each image vertex: it stays the same distance from the mirror as the object vertex, on the opposite side. Drag a vertex onto the mirror line and see that it does not move.
Each image point is the same perpendicular distance from the mirror line as its object point, on the other side.
Vertical mirrors are \(x = k\) (the \(y\)-axis is \(x = 0\)); horizontal mirrors are \(y = k\) (the \(x\)-axis is \(y = 0\)).
Diagonal mirrors (Extended): in \(y = x\) the point \((a, b)\) goes to \((b, a)\); in \(y = -x\) it goes to \((-b, -a)\).
The image is congruent but reversed, like a mirror image of your hand.
Worked example Reflecting in a vertical line
Triangle \(P\) has vertices \((1, 2)\), \((3, 2)\) and \((3, 5)\). Reflect it in the line \(x = 4\).
\(x = 4\) is vertical, so every \(y\)-coordinate stays the same.
\((1, 2)\) is 3 units left of the mirror, so its image is 3 units right: \((7, 2)\).
\((3, 2)\) and \((3, 5)\) are 1 unit left, so their images are 1 unit right: \((5, 2)\) and \((5, 5)\).
Answer: \((7, 2)\), \((5, 2)\), \((5, 5)\)
Worked example Naming the mirror line
Triangle \(A\) has vertices \((2, 1)\), \((4, 1)\), \((4, 3)\). Triangle \(B\) has vertices \((2, -3)\), \((4, -3)\), \((4, -5)\). Describe fully the single transformation that maps \(A\) onto \(B\).
The \(x\)-coordinates are unchanged and the shape is flipped upside down, so it is a reflection in a horizontal line.
The mirror is halfway between corresponding points: \((2, 1)\) and \((2, -3)\) have midpoint \((2, -1)\).
Check with \((4, 3)\) and \((4, -5)\): midpoint \((4, -1)\). Both lie on \(y = -1\).
Answer: Reflection in the line \(y = -1\)
Worked example Diagonal mirror (Extended)
Reflect the point \((4, 1)\) in the line \(y = x\).
Reflecting in \(y = x\) swaps the coordinates.
Check: the midpoint of \((4, 1)\) and \((1, 4)\) is \((2.5, 2.5)\), which is on \(y = x\).
Answer: \((1, 4)\)
Examiner trap
Mixing up \(x = k\) and \(y = k\). The line \(x = 4\) is VERTICAL, so reflecting in it changes the \(x\)-coordinates, not the \(y\)-coordinates. Sketch the mirror line before you reflect anything.
Section 02
Turning about a centre.
In the lab, put the centre of rotation at the origin and turn the shape through \(90^\circ\), \(180^\circ\) and \(270^\circ\). Then move the centre onto a vertex of the shape and repeat: that vertex stays still while everything else swings round it.
Three facts describe a rotation: the angle, the direction, and the centre.
Tracing paper is allowed: trace the shape, pin the centre with your pencil point, turn the paper.
About the origin: \(90^\circ\) clockwise sends \((x, y)\) to \((y, -x)\); \(90^\circ\) anticlockwise sends it to \((-y, x)\); \(180^\circ\) sends it to \((-x, -y)\).
\(180^\circ\) needs no direction: clockwise and anticlockwise give the same image.
Worked example Rotation about the origin
Triangle \(A\) has vertices \((1, 1)\), \((4, 1)\) and \((1, 3)\). Rotate it \(90^\circ\) clockwise about the origin.
\(90^\circ\) clockwise about \((0, 0)\): \((x, y) \to (y, -x)\).
Triangle \(T\) has vertices \((1, 1)\), \((3, 1)\), \((1, 2)\). Rotate it \(90^\circ\) anticlockwise about \((1, 1)\).
The centre \((1, 1)\) is a vertex, so it stays where it is.
\((3, 1)\) is 2 units to the right of the centre. A quarter-turn anticlockwise puts it 2 units above: \((1, 3)\).
\((1, 2)\) is 1 unit above the centre. A quarter-turn anticlockwise puts it 1 unit to the left: \((0, 1)\).
Answer: \((1, 1)\), \((1, 3)\), \((0, 1)\)
Worked example Half-turn about another point (Extended)
Rotate the point \((5, 3)\) through \(180^\circ\) about \((2, 1)\).
From the centre to the point: 3 right and 2 up.
A half-turn reverses this: 3 left and 2 down from the centre.
\((2 - 3, 1 - 2) = (-1, -1)\).
Answer: \((-1, -1)\)
Your turn
Rotate the point \((3, 1)\) through \(90^\circ\) anticlockwise about the origin.
\((x, y) \to (-y, x)\), so \((3, 1) \to (-1, 3)\). Check: the point was on the right, slightly up; a quarter-turn anticlockwise takes it to the top, slightly left.
Examiner trap
Turning the wrong way. Clockwise is the way clock hands move. A \(90^\circ\) clockwise turn is the same as a \(270^\circ\) anticlockwise turn, not a \(90^\circ\) anticlockwise one.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
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What is the image of \((3, 5)\) after reflection in the \(x\)-axis?
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Which describes the mirror line \(x = 2\)?
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What is the image of \((2, -4)\) after a rotation of \(180^\circ\) about the origin?
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What must you state to describe a rotation fully?
Section 03
Describe it fully.
In the lab, read the full description of each transformation. Then, in the exam, decide first whether the image is flipped (a reflection) or just turned (a rotation), and give every fact that the transformation needs.
Worked example A half-turn
Shape \(A\) has vertices \((1, 1)\), \((3, 1)\), \((3, 2)\). Shape \(B\) has vertices \((-1, -1)\), \((-3, -1)\), \((-3, -2)\). Describe fully the single transformation that maps \(A\) onto \(B\).
Every coordinate has changed sign: \((x, y) \to (-x, -y)\).
That is the rule for a half-turn about the origin.
The shape is not flipped (its vertices go round in the same order), so it is a rotation, not a reflection.
Answer: Rotation, \(180^\circ\), about \((0, 0)\)
Worked example Finding the centre (Extended)
Triangle \(P\) has vertices \((1, 0)\), \((3, 0)\), \((3, 1)\). Triangle \(Q\) has vertices \((4, 1)\), \((4, 3)\), \((3, 3)\). Describe fully the single transformation that maps \(P\) onto \(Q\).
The side from \((1, 0)\) to \((3, 0)\) points right; its image from \((4, 1)\) to \((4, 3)\) points up. A quarter-turn anticlockwise.
The centre is the same distance from each point as from its image. Try \((2, 2)\): \((1, 0)\) is 1 left and 2 down from it. A quarter-turn anticlockwise turns '1 left, 2 down' into '2 right, 1 down', which lands on \((4, 1)\), the image.
Check another vertex: \((3, 1)\) is 1 right and 1 down from \((2, 2)\); turned anticlockwise it becomes 1 right and 1 up, landing on \((3, 3)\). Correct.
Answer: Rotation, \(90^\circ\) anticlockwise, about \((2, 2)\)
Examiner trap
Naming two transformations, such as 'a reflection and then a translation', or leaving out the centre or direction. The question asks for a SINGLE transformation, and a rotation without its centre scores very little.
Stage 2 · Exam
Exam-style questions
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A triangle has a vertex at \((5, 2)\). The triangle is reflected in the line \(x = 3\). Where is the image of this vertex?
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The point \((4, 1)\) is rotated \(90^\circ\) clockwise about the origin. What are the coordinates of its image?
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(Extended) The point \((5, -1)\) is reflected in the line \(y = x\). What are the coordinates of its image?
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(Extended) The point \((4, 3)\) is rotated \(90^\circ\) anticlockwise about \((1, 2)\). What are the coordinates of its image?
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