Coordinate Plane:Rotations

A rotation is a transformation that “turns” a figure about a point in either a clockwise or counterclockwise direction.

The point about which the figure rotates is called the center of rotation.

A rotation image is congruent to original figure and have different orientations.

To describe a rotation, tell whether the turn is clockwise or counterclockwise and the number of degrees the figure is turned around its center of rotation.

A 90° rotation is a quarter turn.A 180° rotation is a half turn.A 270° rotation is a three-quarter turn.

Rules for Counterclockwise Rotation About the Origin90° rotation: (x,y)

180° rotation: (x,y)

270° rotation: (x,y)

(-y, x)

(-x, -y)

(y, -x)

Rules for Clockwise Rotation About the Origin90° rotation: (x,y)

180° rotation: (x,y)

270° rotation: (x,y) (-y, x)

(-x, -y)

(y, -x)

You can draw a rotation of a point P(x,y) counterclockwise about the origin on a coordinate plane.

ExampleRotate P(-2,3) 90°, 180°, and 270° counterclockwise about the origin.

● 90° rotation:(x, y) → (-y, x)P(-2, 3) → P′ (-3, -2)

Now graph the point.

Change the sign of the y-coordinate, and then reverse the order of the coordinates.

● 180° rotation:(x, y) → (-x, -y)P(-2, 3) → P′′ (2, -3)

Now graph the point.

Change the sign of both the x- and y-coordinates.

● 270° rotation:(x, y) → (y, -x)P(-2, 3) → P′′′ (3, 2)

Now graph the point.

Change the sign of the x-coordinate, and then reverse the order of the coordinates.

To rotate a polygon 90°, 180°, or 270° about the origin, rotate the vertices, and then connect their images to form the image of the polygon.

ExampleRotate ABC with vertices A(-5,-2), B(-1,-2), and C(-4,-4) 90° counterclockwise about the origin.

A( -5, -2) →

B(-1, -2) →

C (-4, -4) →

90° rotation: (x,y) → (-y,x)A′ (2, -5)

B′ (2, -1)

C′ (4, -4)

Now graph the points and connect for form the triange.

Segments from the origin to a point on the original polygon and the origin to the corresponding point on the rotation image form a 90° angle.

You Try!Graph the triangle with the vertices H(-5, 1), I(1, 4), J(2, 2) and its rotation image about the origin on the same coordinate grid. 1. 90°; counterclockwise2. 180°; clockwise3. 270°; counterclockwise

Answer Keys:1. H′(-1, - 5); I′(-4, 1); J′(-2,2)2. H′′(5,-1); I′′(-1,-4); J′′(-2,-2)3. H′′′(1,5); I′′′(4,-1); J′′′(2,-2)