A center, an angle and a direction
A rotation turns a shape about one fixed point, called the center of rotation. Every point of the shape turns through the same angle, in the same direction, either counterclockwise or clockwise. A quarter turn is 90°, a half turn is 180°, and a full turn is 360°.
To describe a rotation fully, give all three: the center, the angle and the direction. A rotation, like a reflection, keeps every length and every angle, so the image is congruent to the object. Unlike a reflection, it does not turn the shape over: the corners still go round in the same order.
A quarter turn counterclockwise about the origin, the gold dot. The triangle A has corners (1, 1), (4, 1) and (4, 2); its image A' has corners (−1, 1), (−1, 4) and (−2, 4).
Every point circles the center
As the shape turns, each point moves along a circle whose center is the center of rotation. So each point ends exactly as far from the center as it started. The corner (4, 2) is from the origin, and its image (−2, 4) is from the origin too.
The line from the center to a point turns through the angle of the rotation, here 90°, to become the line from the center to its image. Only one point does not move: the center itself.
A quarter turn counterclockwise about the origin: (2, 1) goes to (−1, 2), (4, 2) goes to (−2, 4), and (2, 3) goes to (−3, 2). Each image is as far from the origin as its corner.
Why a quarter turn sends (x, y) to (−y, x)
Take the point (4, 1). To reach it from the origin, go 4 along the x-axis to (4, 0), then 1 up. Those two moves and the line back to the origin make a right triangle, with corners (0, 0), (4, 0) and (4, 1).
Turn the whole triangle a quarter turn counterclockwise about the origin. The side that ran 4 to the right now runs 4 up, along the y-axis. The side that ran 1 up now runs 1 to the left. So the tip of the triangle, the image of (4, 1), is 1 to the left and 4 up: (−1, 4).
The same happens to every point: its x-coordinate becomes the distance up, and its y-coordinate becomes the distance to the left. So a quarter turn counterclockwise about the origin sends (x, y) to (−y, x). The coordinates swap, and then the new x-coordinate changes sign. For (2, 1): swap to (1, 2), then change the sign of the first, giving (−1, 2). A negative coordinate is a move the other way, and it turns in the same way, so the rule holds in every quadrant: (−3, 2) goes to (−2, −3).
The right triangle from the origin to (4, 1), with sides 4 across and 1 up, and its image, dashed, after a quarter turn counterclockwise. The side of 4 now runs up the y-axis and the side of 1 runs to the left, so the tip is at (−1, 4).
A half turn, and a clockwise quarter turn
A half turn is two quarter turns. Apply the rule twice: . So a half turn about the origin changes both signs: (4, 2) goes to (−4, −2). Turning 180° clockwise ends in exactly the same place as turning 180° counterclockwise, so a half turn needs no direction.
A point and its image after a half turn are on a straight line through the center, one on each side, and the center is the midpoint between them.
A quarter turn clockwise is three quarter turns counterclockwise, 270°. Apply the rule three times: . So a quarter turn clockwise sends (2, 1) to (1, −2).
A half turn about the origin: (1, 1), (4, 1) and (4, 2) go to (−1, −1), (−4, −1) and (−4, −2). Both signs change.
Finding the center
When the center is not given, find it from a shape and its image. The center is as far from a point as it is from that point's image, because the point moves along a circle about the center. So the center lies on the perpendicular bisector of the segment joining any point to its image.
Take A(5, 1) going to A'(2, 4), and B(5, 3) going to B'(0, 4). The midpoint of AA′ is (3.5, 2.5), and AA′ has gradient , so its perpendicular bisector has gradient 1: y − 2.5 = x − 3.5, which is y = x − 1.
The midpoint of BB′ is (2.5, 3.5), and BB′ has gradient , so its perpendicular bisector has gradient 5: y − 3.5 = 5(x − 2.5), which is y = 5x − 9.
The center is on both lines: x − 1 = 5x − 9, so 4x = 8, x = 2 and y = 1. The center is (2, 1). From it, A is 3 to the right and A' is 3 up, so the rotation is 90° counterclockwise. Check with B: from the center it is 3 to the right and 2 up; a quarter turn counterclockwise makes that 2 to the left and 3 up, which reaches (0, 4), where B' is.
The gold line is the perpendicular bisector of A(5, 1) and A'(2, 4); the white line is the perpendicular bisector of B(5, 3) and B'(0, 4). They cross at the center, (2, 1). The dashed circle about the center passes through both A and A'.
The usual mistakes
Turning the wrong way. A quarter turn counterclockwise sends (2, 1) to (−1, 2); (1, −2) is where a clockwise quarter turn sends it.
Confusing the quarter turn and the half turn. (−2, −1) is the half turn of (2, 1), twice as far round as 90°.
Swapping without the sign change, or changing a sign without swapping. (1, 2) is the reflection of (2, 1) in y = x, and (−2, 1) is its reflection in the y-axis; neither is a rotation of it.
Taking the midpoint of a point and its image as the center. The center is somewhere on their perpendicular bisector, not necessarily at the midpoint; a second pair of points fixes it.
Leaving out part of the description. "A turn of 90°" could be clockwise or counterclockwise, about any point: name the center, the angle and the direction.
Steps written as column vectors
A step across and up, such as "3 to the right and 1 down", can be written as a column vector: the two numbers are stacked in brackets, the step across on top and the step up underneath. Across is positive to the right and negative to the left; up is positive upward and negative downward. On one line it is written [3, −1].
So from the center (2, 1), the point A(5, 1) is the step [3, 0], and A'(2, 4) is the step [0, 3]. A quarter turn counterclockwise sends the step [a, b] to [−b, a], by the same rule as for points about the origin.
Worked example: A Robot Arm That Swings a Part Across a Workbench: the Pivot and the Turn
Question A robot arm turns about a pivot hidden under its base cover. On a plan of a workbench marked in decimeters, it picks up a triangular part with corners A(6, 1), B(8, 1) and C(6, 2) and puts it down with its corners at A'(4, 5), B'(4, 7) and C'(3, 5). The move is a single rotation about the pivot. (a) Find the coordinates of the pivot, the center of the rotation. (b) Find the angle of the rotation, less than 180°, and its direction.
1.A(6, 1) goes to A'(4, 5). The midpoint of AA' is (5, 3), and AA' has gradient 5 − 14 − 6 = −2, so its perpendicular bisector has gradient 12 and equation y = 3 + 12(x − 5).
The pivot is as far from A as from A', so it lies on the perpendicular bisector of AA': y = 3 + 12(x − 5). 2.B(8, 1) goes to B'(4, 7). The midpoint of BB' is (6, 4), and BB' has gradient 7 − 14 − 8 = −32, so its perpendicular bisector has gradient 23 and equation y − 4 = 23(x − 6), which is y = 23x.
It also lies on the perpendicular bisector of BB': y = 23x. 3.(a) The bisectors meet where 23x = 3 + 12(x − 5). Multiplying by 6 gives 4x = 18 + 3x − 15, so x = 3 and y = 23 × 3 = 2. The pivot is at (3, 2).
(a) The two bisectors meet at (3, 2): that is the pivot. 4.From the pivot, A is 3−1 away and A' is 13 away. Both have length √10, and their gradients, −13 and 3, multiply to −1, so the lines from the pivot to A and to A' are at right angles.
From the pivot, A is 3−1 away and A' is 13 away: equal lengths, at right angles. 5.(b) The direction to A points right and a little down, and the direction to A' points up and a little right, so the turn from one to the other is a quarter turn counterclockwise. The rotation is 90° counterclockwise about (3, 2). Check: C is 30 from the pivot, and a quarter turn counterclockwise makes that 03, which reaches (3, 5), where C' is.
(b) The rotation is 90° counterclockwise about (3, 2); it takes C to (3, 5), which is C'.
Answer: (a) the pivot is at (3, 2); (b) 90° counterclockwise
Common mistakes
- Taking the midpoint of AA', (5, 3), as the center. The center lies on the perpendicular bisector of AA' but may be anywhere along it; a second pair of points is needed to fix it.
- Giving the angle without its direction. A quarter turn clockwise about (3, 2) would send A to (2, −1), nowhere near where the part was put down, so the direction is part of the answer.