SECTION 9.3Rotations

In Lesson 4.7, you learned that a rotation or turn moves every point of a preimage through a specified angle and direction about a fixed point.

The direction of rotation can be either clockwise or counterclockwise. Assume that all rotations are counterclockwise unless stated otherwise.

• Draw a segment from point R to point A.

• Locate point R' so that AR = AR'.

Example 1: Rotate quadrilateral RSTV 45° counterclockwise about point A.

• Repeat this process for points S, T, and V.

• Connect the four points to form R'S'T'V'.

• Use a protractor to measure a 45° angle counterclockwise with as one side. Extend the other side to be longer than AR.

AR

Quadrilateral R'S'T'V' is the image of quadrilateral RSTV under a 45° counterclockwise rotation about point A.

When a point is rotated 90°, 180°, or 270° counterclockwise about the origin, you can use the following rules:

Example 2: Triangle DEF has vertices D(–2, –1), E(–1, 1), and F(1, –1). Graph ΔDEF and its image after a rotation of 115° clockwise about the point G(–4, –2).

First, draw ΔDEF and plot point G.

Repeat with points E and F.

Draw a segment from point G to point D.

Use a protractor to measure a 115° angle clockwise with as one side.GD

Draw .GR

Use a compass to copy onto Name the segment

GD .GR'.GD

ΔD'E'F' is the image of ΔDEF under a 115° clockwise rotation about point G.

Example 3: Hexagon DGJTSR is shown below. What is the image of point T after a 90 counterclockwise rotation about the origin?

Multiple Choice:

a) (5, –3)

b) (–5, –3)

c) (–3, 5)

d) (3, –5)

Explanation on next slide!

Read the Test Item

You are given a graph of hexagon DGJTSR and asked to identify the coordinates of the image of point T after a 90° counterclockwise rotation about the origin.

Solve the Test Item

To find the coordinates of point T after a 90 counterclockwise rotation about the origin, multiply the y-coordinate by –1 and then interchange the x- and y-coordinates.

(x, y) → (–y, x) (5, 3) → (–3, 5)

Answer: The answer is C, (–3, 5).

Example 4: Triangle PQR is shown below. What is the image of point Q after a 90° counterclockwise rotation about the origin?

To find the coordinates of point Q after a 90 counterclockwise rotation about the origin, multiply the y-coordinate by –1 and then interchange the x- and y-coordinates.

(x, y) → (–y, x) (4, 5) → (–5, 4)