Rotation (Geometry)
In short: Rotation swings all points of a shape through one common angle around a pinned center, so size and form survive while the shape ends up pointing a new way.
8th GradeGeometry8.G.A.1
What is Rotation?
Pin a cardboard shape at one spot and spin it: that is a rotation about a center, through a stated angle and direction. Each point holds its distance from the pin and travels along a circle, which is why lengths and angles stay unchanged.
About the origin on a coordinate plane, a \(90^{\circ}\) turn counterclockwise sends \((x, y)\) to \((-y, x)\); a \(180^{\circ}\) turn sends it to \((-x, -y)\); a \(90^{\circ}\) turn clockwise sends it to \((y, -x)\). Counterclockwise is the positive direction unless told otherwise.
How to spot it
- The wording says turn, spin or rotate a shape by 90, 180 or 270 degrees about a point.
- A pivot, usually the origin, is named or marked.
- A picture shows the original and a tilted copy of identical shape.
- A design has wheel-like symmetry, such as a pinwheel or a starburst.
The method
- Pin down the pivot, the angle and the direction, clockwise or counterclockwise.
- Run each corner through the coordinate rule for that angle about the origin.
- Mark the new corners and join them in the original order.
- Test the result: every corner should sit as far from the pivot as it did before.
Rotation: solved example
Problem
A paper triangle ABC sits at \(A(3, 1)\), \(B(6, 1)\) and \(C(3, 4)\). Spin it \(90^{\circ}\) counterclockwise around the origin and report where it lands.
Solution
Apply the quarter-turn rule \((x, y) \to (-y, x)\) to each corner.
\(A^{\prime} = (-1, 3)\), \(B^{\prime} = (-1, 6)\), \(C^{\prime} = (-4, 3)\). Both legs still measure 3, so the new triangle has corners \((-1, 3)\), \((-1, 6)\) and \((-4, 3)\), the same right triangle after a quarter turn.
Try it
Question 1
Rotate the point \((5, -2)\) by \(180^{\circ}\) about the origin.
Show the answer
\((x, y) \to (-x, -y)\) gives \((-5, 2)\).
Question 2
Rotate \((4, 7)\) by \(90^{\circ}\) clockwise about the origin.
Show the answer
The clockwise rule is \((x, y) \to (y, -x)\), so the image is \((7, -4)\).
Rotation vs. Reflection
A rotation keeps the figure’s orientation and turns it about a point through an angle. A reflection reverses orientation and flips it across a line. Doing two reflections over crossing lines produces a rotation, which shows how closely the two are related.
Keep going
Lessons
Worksheets and quizzes
- Translations, Reflections and Rotations – Worksheet
- Translations, Reflections and Rotations – Quiz
- Translations, Reflections and Rotations – Worksheet
- Translations, Reflections and Rotations – Quiz
Related words
Whole dictionary: A–Z · 8th Grade · Geometry