A rotation is a transformation that turns a figure around a fixed point, called the center, by a given angle and direction. Every point stays the same distance from the center, so size and shape do not change. About the origin, a 90° counterclockwise turn sends (x, y) to (−y, x).
Each vertex moves along a circle centered at the origin, so it stays the same distance from the center while it turns 90°.
Formal definition and notation
A rotation about a point O through an angle θ maps O to itself and maps every other point P to the point P′ with OP′ = OP and ∠ POP′ = θ, measured counterclockwise for positive θ and clockwise for negative θ. Rotations are rigid motions: they preserve distances, angle measures, parallel lines and the orientation (clockwise or counterclockwise order) of vertices. A rotation of 360° returns every point to its start.
Symbol
Read it as
Meaning
RO, 90°
rotation about O by 90 degrees
a common way to name a rotation by its center and angle
A′
A prime
the image of point A after the rotation
Key facts
A rotation needs three facts: the center, the angle and the direction (clockwise or counterclockwise). If no direction is given, counterclockwise is assumed.
A 180° rotation is the same clockwise or counterclockwise, and it sends (x, y) to (−x, −y).
Rotations preserve sidelengths, angle measures and area, so the image is congruent to the original.
The center of rotation is the only point that does not move (unless the angle is a multiple of 360°).
A shape has rotational symmetry if a turn of less than 360° maps it onto itself: a square every 90°, an equilateral triangle every 120°.
A reflection flips a figure over a line and reverses its orientation; a rotation turns it around a point and keeps orientation. Reflecting (3, 2) over y = x gives (2, 3); rotating it 90° gives (−2, 3).
A translation slides every point the same distance in the same direction; a rotation turns points by different distances depending on how far they are from the center.
A dilation changes size by a scale factor; a rotation never changes size.
Quick check: 3 questions
Question 1 of 3
What is the image of (5, −2) after a 180° rotation about the origin?
Show answer and explanation
Answer: A. A 180° rotation changes both signs: (−5, 2). (5, 2) is a reflection over the x-axis, and the others swap the coordinates as a 90° turn would.
Question 2 of 3
Rotate (1, 4) 90° counterclockwise about the origin.
Show answer and explanation
Answer: C. Use (x, y) → (−y, x): (−4, 1). (4, −1) is the clockwise turn, (−1, 4) is a reflection over the y-axis, and (−4, −1) negates both after swapping.
Question 3 of 3
Which of these can change when a figure is rotated?
Show answer and explanation
Answer: D. A rotation is a rigid motion, so lengths, angles and area stay the same. Only the position, and so the coordinates, change.
Rotation: frequently asked questions
What are the rules for rotations on a coordinate plane?
About the origin: 90° counterclockwise, (x, y) → (−y, x); 180°, (x, y) → (−x, −y); 270° counterclockwise (90° clockwise), (x, y) → (y, −x). Test with (2, 5): the images are (−5, 2), (−2, −5) and (5, −2).
Is a rotation clockwise or counterclockwise?
Either, so a problem should say which. When it gives only an angle, math uses counterclockwise as the positive direction, so “rotate 90°” means counterclockwise. A clockwise rotation of 90° lands in the same place as a counterclockwise rotation of 270°.
Does a rotation make congruent figures?
Yes. A rotation is a rigid motion: every distance and every angle stays the same, so the image is congruent to the original. It also keeps orientation, meaning the vertices still go around in the same clockwise or counterclockwise order, unlike a reflection.
TransformationA transformation moves or changes a figure on the coordinate plane. The four common transformations are translations (slides), reflections (flips), rotations (turns) and dilations (resizing). The original figure is the preimage and the result is the image, labeled with primes such as A′. Only a dilation changes the size.Geometry · Grade 7, Grade 8, Algebra 1Parallel LinesParallel lines are lines in the same plane that never intersect. They stay the same distance apart and point in the same direction, like the rails of a straight railroad track. On a coordinate plane, parallel lines have the same slope but different y-intercepts. The symbol ∥ means is parallel to.Geometry · Grade 4, Grade 5, Grade 6SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2LengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3CongruentCongruent means exactly the same size and shape. Two segments are congruent when they have the same length, two angles when they have the same measure, and two figures when one can be turned, flipped or slid onto the other. The symbol is ≅: AB ≅ CD reads “segment AB is congruent to segment CD.”Geometry · Grade 8, GeometrySymmetrySymmetry is when a shape matches itself after it is flipped or turned. A shape has line symmetry when a line splits it into two mirror-image halves that match exactly when folded; that fold line is the line of symmetry. It has rotational symmetry when it looks the same after a turn of less than 360°.Geometry · Grade 4, Grade 8, GeometrySquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3Equilateral TriangleAn equilateral triangle is a triangle whose three sides all have the same length. Its three angles are also equal, so each one measures 60 degrees. With side length a, its perimeter is 3a and its area is √34a2. Every equilateral triangle is also isosceles and acute.Geometry · Grade 4, Grade 5, Grade 7OriginThe origin is the point where the x-axis and the y-axis cross on the coordinate plane. Its coordinates are (0, 0), and it is labeled O. Every ordered pair tells how far to move from the origin: (4, 3) means 4 units right, then 3 units up.Geometry · Grade 5, Grade 6, Grade 8TriangleA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).Geometry · Grade 3, Grade 4, Grade 5SimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 1824 = 34. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5, Grade 6ReflectionA reflection is a transformation that flips a figure over a line to make a mirror image. That line is the line of reflection: each point and its image are the same distance from the line, on opposite sides. The image is congruent but faces the other way. Across the y-axis, (x, y) goes to (−x, y).Geometry · Grade 8, GeometryQuadrantA quadrant is one of the four regions that the x-axis and y-axis split the coordinate plane into. They are numbered I, II, III and IV counterclockwise, starting at the top right. The signs of a point’s coordinates tell its quadrant: (+, +) is I, (−, +) is II, (−, −) is III and (+, −) is IV.Geometry · Grade 5, Grade 6, Grade 7TranslationA translation is a slide: a transformation that moves every point of a figure the same distance in the same direction, without turning or flipping it. The image is congruent to the original. On the coordinate plane, a translation adds the same amounts to every point: (x, y) → (x + a, y + b).Geometry · Grade 8, GeometryDilationA dilation is a transformation that enlarges or shrinks a figure by a scale factor k from a fixed center point. Each point moves along the ray from the center to k times its distance. With center at the origin, (x, y) → (kx, ky). The image keeps its shape, so it is similar, not congruent.Geometry · Grade 8, GeometryScale FactorA scale factor is the number used to multiply side lengths when making a similar figure larger or smaller. A scale factor of 2 doubles every length, and a scale factor of 12 makes lengths half as large. Find it by dividing a new length by the matching original length. Areas change by the square of the scale factor.Ratios and Proportions · Grade 7, Grade 8, Algebra 1Coordinate PlaneThe coordinate plane is a flat grid formed by a horizontal x-axis and a vertical y-axis crossing at the origin, (0, 0). Every point is located by an ordered pair, x first then y, such as (3, −2). The two axes divide the plane into four quadrants, numbered I to IV counterclockwise from the upper right.Geometry · Grade 5, Grade 6, Grade 7AngleAn angle is formed by two rays that share the same endpoint, called the vertex. It is measured in degrees by how far one ray is turned from the other. A right angle measures 90 degrees, a straight angle measures 180 degrees, and one full turn is 360 degrees.Geometry · Grade 4, Grade 5, Grade 6Ordered PairAn ordered pair is two numbers written to locate a point on the coordinate plane. It is written as (x, y): the x-value comes first, and the y-value comes second. From the origin, move x units across, then y units up or down. The order matters: (2, 5) and (5, 2) are different points.Geometry · Grade 5, Grade 6, Grade 7Congruent FiguresCongruent figures have exactly the same size and shape. One can be moved onto the other by sliding, flipping or turning, so every pair of matching sides and angles is equal. Two squares with side 5 are congruent; a 3-4-5 triangle and a 6-8-10 triangle are not, because their sizes differ. The symbol is ≅.Geometry · Grade 7, Grade 8