Key takeaway: A rotation turns every point of a figure through the same angle around a fixed point called the center of rotation, keeping the size and shape unchanged.
Grade 88.G.A.1Geometry and Measurement
Definition of Rotation
A rotation is a turn. Imagine pinning a paper triangle at one point and swinging it around that pin: each vertex travels along a circle centered at the pin, and all of them sweep through the same angle. The pin is the center of rotation.
Two facts describe it: the angle and the direction. Counterclockwise is the positive direction in math, and clockwise is negative. Distances from the center stay the same, so the image is congruent to the original. If the center is the origin, simple coordinate rules exist for quarter turns and half turns. A 180 degree turn sends each point straight through the center to the opposite side.
Formula and notation
About the origin, a counterclockwise quarter turn follows
\[90^{\circ}\text{ CCW: } (x, y) \to (-y, x)\]
and a half turn follows \(180^{\circ}: (x, y) \to (-x, -y)\).
Key facts
- Each point stays the same distance from the center during the turn.
- Counterclockwise is positive; clockwise is negative.
- A 360 degree turn returns every point to where it began.
- Lengths and angle measures do not change.
Rotation example
Rotate triangle \(ABC\) with \(A(4, 1)\), \(B(7, 1)\) and \(C(7, 3)\) by \(90^{\circ}\) counterclockwise about the origin. Find the image vertices.
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Use the rule \((x, y) \to (-y, x)\).
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\(A(4, 1) \to A^{\prime}(-1, 4)\) and \(B(7, 1) \to B^{\prime}(-1, 7)\).
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\(C(7, 3) \to C^{\prime}(-3, 7)\).
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Check: \(AB = 3\) (horizontal) and \(A^{\prime}B^{\prime} = 3\) (vertical). The lengths match.
The image vertices are \(A^{\prime}(-1, 4)\), \(B^{\prime}(-1, 7)\), \(C^{\prime}(-3, 7)\).
Common mistake
Watch out
For the same turn, Faye uses \((x, y) \to (y, -x)\) and gets \(A^{\prime}(1, -4)\).
Why it is wrong
That rule is for a quarter turn clockwise. The problem says counterclockwise, so use \((-y, x)\) and get \(A^{\prime}(-1, 4)\). Sketch the point: \(A\) sits in the first quadrant, and a counterclockwise quarter turn must land it in the second.
Related terms and lessons
- Grade 8 · Translations, Reflections and RotationsLessonWorksheetQuiz
- Geometry · Rigid TransformationsLessonWorksheetQuiz
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