Rotations About the Origin
A rotation needs three things: a centre, an angle and a direction. For quarter turns about the origin the coordinate rules are short, and each one can be rebuilt from the picture rather than memorised in isolation.
Plot $(4, 0)$ and turn it a quarter turn anticlockwise about the origin. Where does it land? Now do the same for $(0, 3)$. What is happening to the pair of numbers?
A quarter turn anticlockwise sends $(x, y)$ to $(-y, x)$: swap, then negate the new x. A half turn sends it to $(-x, -y)$. A quarter turn clockwise sends it to $(y, -x)$: swap, then negate the new y.
$$(x, y) \to (-y, x) \to (-x, -y) \to (y, -x)$$
Every one of these rules swaps the coordinates and changes a sign, except the half turn, which changes both signs and swaps nothing. If you can remember where $(1, 0)$ goes, you can rebuild any of them in seconds.
Know
- that a rotation of $90°$ anticlockwise about the origin sends $P(x, y)$ to $P'(-y, x)$
- that a rotation of $180°$ about the origin sends $P(x, y)$ to $P'(-x, -y)$, in either direction
- that a rotation of $90°$ clockwise, equivalently $270°$ anticlockwise, sends $P(x, y)$ to $P'(y, -x)$
Understand
- why a rotation about the origin leaves the distance from the origin unchanged
- why the origin is the only invariant point of a rotation about the origin, other than for the full turn
- why a rotation preserves orientation while a reflection reverses it
Can Do
- find the image of a point or shape under a rotation of a multiple of $90°$ about the origin
- convert between clockwise and anticlockwise descriptions of the same rotation
- identify which rotation maps a given point onto a given image, and check it using distances
A rotation is not fully described until three things are given: the centre, the angle and the direction.
In this lesson the centre is always the origin and the angle is always a multiple of $90°$, which keeps the arithmetic to sign changes. The direction still has to be stated, except for a half turn, where clockwise and anticlockwise give the same result.
The convention matches the unit circle: anticlockwise is positive. A question saying "rotate $90°$ about the origin" without further qualification means anticlockwise.
A rotation preserves lengths, angles and orientation. The image is congruent and faces the same rotational way round, which is what separates a rotation from a reflection.
Rather than memorising the rule, watch two easy points.
The point $(1, 0)$ sits on the positive x-axis. A quarter turn anticlockwise takes it to the positive y-axis, at $(0, 1)$.
The point $(0, 1)$ sits on the positive y-axis. The same quarter turn takes it to the negative x-axis, at $(-1, 0)$.
So the x-direction becomes the y-direction, and the y-direction becomes the negative x-direction. A general point $(x, y)$ is $x$ steps in the first direction and $y$ steps in the second, so its image is $x$ steps up and $y$ steps left:
$P(x, y) \to P'(-y, x)$
Two points you can picture generate the rule, which is more reliable than a memorised string of symbols with a sign that may have drifted.
Apply the quarter turn twice to get the half turn:
$(x, y) \to (-y, x) \to (-x, -y)$
So a rotation of $180°$ negates both coordinates. Turning the other way gives the same answer, which is why direction need not be stated for a half turn.
Apply it three times for the three-quarter turn:
$(-x, -y) \to (y, -x)$
A rotation of $270°$ anticlockwise is the same as $90°$ clockwise, so both are described by $(x, y) \to (y, -x)$.
Applying it a fourth time returns $(x, y)$, which confirms the chain closes correctly. If your four rules do not return you to the start, one of them has a sign wrong.
Any clockwise rotation can be renamed as an anticlockwise one by subtracting from $360°$.
$90°$ clockwise is $270°$ anticlockwise. $270°$ clockwise is $90°$ anticlockwise. $180°$ in either direction is the same transformation.
This matters because a question may state one and an answer key the other, and they are not disagreeing. Converting to a single convention before comparing avoids a false alarm.
The two quarter turns are easy to confuse, and swapping them is the usual error. The check takes seconds: track $(1, 0)$. Anticlockwise sends it up to $(0, 1)$; clockwise sends it down to $(0, -1)$.
Two checks catch nearly every mistake.
The distance from the origin cannot change. Rotating $(4, 1)$ gives $(-1, 4)$, and both are $\sqrt{17}$ from the origin. If your image is a different distance away, the rule was applied wrongly.
For a quarter turn, the two radii are perpendicular. The gradient of $OP$ is $\tfrac{1}{4}$ and the gradient of $OP'$ is $\tfrac{4}{-1} = -4$, and their product is $-1$. This is the same negative-reciprocal fact from Lesson 8, appearing here as a consequence of turning through a right angle.
Notice also that the origin is the only point that does not move, since every other point is at a non-zero distance and gets carried round a circle. A rotation has exactly one invariant point, a reflection has a whole line of them, and a translation has none.
Watch Me Solve It · 3 examples
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1Apply the rule$(x, y) \to (-y, x)$$A(4, 1) \to A'(-1, 4)$Swap the coordinates to get $(1, 4)$, then negate the new first coordinate.
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2Check the distance from the origin$OA = \sqrt{16 + 1} = \sqrt{17}$$OA' = \sqrt{1 + 16} = \sqrt{17}$Equal, as a rotation about the origin requires.
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3Check the right angle$m_{OA} = \tfrac{1}{4}, \qquad m_{OA'} = \tfrac{4}{-1} = -4$$\tfrac{1}{4} \times (-4) = -1$The two radii are perpendicular, which confirms the turn was a quarter and not something else.
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4Check the direction$A$ is in the first quadrant and $A'$ is in the second, which is the anticlockwise direction. Clockwise would have sent it to the fourth quadrant.
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1State the rule$(x, y) \to (-x, -y)$Both coordinates change sign. The direction need not be specified for a half turn.
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2Apply it to each vertex$P(2, 3) \to P'(-2, -3)$$Q(-1, 5) \to Q'(1, -5)$$R(4, -2) \to R'(-4, 2)$Take care with the vertices that already have a negative coordinate.
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3Check one distance$OQ = \sqrt{1 + 25} = \sqrt{26}$$OQ' = \sqrt{1 + 25} = \sqrt{26}$Unchanged, as required.
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4Note what is preservedThe image is congruent to the original and has the same orientation, since a rotation does not reverse it. Only the origin is invariant, and no vertex is at the origin here.
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1Recall the clockwise rule$(x, y) \to (y, -x)$Swap the coordinates, then negate the new second one. Checking on $(1, 0)$: it maps to $(0, -1)$, which is downwards, so the rule is indeed clockwise.
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2Apply it$B(-3, 5) \to B'(5, 3)$Swapping gives $(5, -3)$, and negating the new second coordinate gives $(5, 3)$.
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3Check the distance$OB = \sqrt{9 + 25} = \sqrt{34}$$OB' = \sqrt{25 + 9} = \sqrt{34}$Equal, as required.
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4Give the equivalent$360° - 90° = 270°$A rotation of $90°$ clockwise is the same transformation as $270°$ anticlockwise.
Brain Trainer · 5 problems
Five items. All rotations are about the origin, and anticlockwise unless stated otherwise.
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1 Rotate $(2, 7)$ by $90°$ anticlockwise.
$(x, y) \to (-y, x)$.$(-7, 2)$ -
2 Rotate $(2, 7)$ by $180°$.
Negate both coordinates.$(-2, -7)$ -
3 Rotate $(2, 7)$ by $90°$ clockwise.
$(x, y) \to (y, -x)$.$(7, -2)$ -
4 Rotate $(-5, -1)$ by $90°$ anticlockwise.
Swap to $(-1, -5)$, then negate the first.$(1, -5)$ -
5 Which point is invariant under a rotation of $90°$ about the origin?
Every other point is carried round a circle of non-zero radius.Only the origin
Multiple Choice · 5 questions
Rotated $90°$ anticlockwise about the origin, the point $(6, 2)$ maps to:
A rotation about the origin maps $(3, -8)$ to $(-3, 8)$. It is:
A rotation of $270°$ clockwise about the origin is the same as:
A student rotates $(5, 2)$ about the origin and obtains $(2, 5)$. This must be wrong as a quarter turn because:
A rotation of $90°$ about the origin has:
Short Answer · 3 questions
Three rules, all swaps and signs
$90°$ anticlockwise gives $(-y, x)$; $180°$ gives $(-x, -y)$; $90°$ clockwise, or $270°$ anticlockwise, gives $(y, -x)$.
Rebuild rather than memorise
Track $(1, 0)$ and $(0, 1)$ through the turn. Anticlockwise sends $(1, 0)$ up to $(0, 1)$; clockwise sends it down to $(0, -1)$.
Two checks catch most errors
The distance from the origin must be unchanged, and for a quarter turn the two radii must have gradients multiplying to $-1$.
Invariant points identify the type
A translation has none, a rotation has exactly one, and a reflection has a whole line of them. Counting them names the transformation.
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