Lesson 5Definition of Rotation and

Basic Properties

BASIC RIGID MOTION #3 = Rotation

when a figure is moved around a fixed point a specific number of degrees

rotation symbol: R

Think: A skateboarder doing a 180 or a 360Intuitively --- “turn”

Center of Rotation : The fixed point from which a figure is turned or rotated.

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NEED TO KNOWnumber of degrees AND direction

NAME THE POSITIVE OR NEGATIVE NUMBER OF

DEGREES THE FIGURE IS TO BE ROTATED BY

AND

NAME WHETHER THE ROTATION IS CLOCKWISE or COUNTERCLOCKWISE

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Clockwise- the direction the hands on the clock move; when rotating a figure clockwise (CW) the number of degrees is negative Counter-clockwise- the opposite direction the hands on the clock move; when rotating a figure counter-clockwise (CCW) the number of degrees is positive.

A positive degree of rotation moves the figure counterclockwise. (CCW)

A negative degree of rotation moves the figure clockwise. (CW)

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LESSON 5: DEFINITION of ROTATIONS

1. Let there be a rotation of 𝑑 degrees around center 𝑂. Let 𝑃 be a point other than 𝑂. Select 𝑑 so that 𝑑≥ 0. Find 𝑃′ (i.e., the rotation of point 𝑃) using a transparency.

2. Let there be a rotation of 𝑑 degrees around center 𝑂. Let 𝑃 be a point other than 𝑂. Select 𝑑 so that 𝑑< 0. Find 𝑃′ (i.e., the rotation of point 𝑃) using a transparency.

3. Which direction did the point 𝑃 rotate when 𝑑≥ 0?

4. Which direction did the point 𝑃 rotate when 𝑑< 0?

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5. Let 𝐿 be a line, 𝐴𝐵ሬሬሬሬሬԦ be a ray, 𝐶𝐷 be a segment, and ∠𝐸𝐹𝐺 be an angle, as shown. Let there be a rotation of 𝑑 degrees around point 𝑂. Find the images of all figures when 𝑑≥ 0.

6. Let 𝐴𝐵തതതത be a segment of length 4 units and ∠𝐶𝐷𝐸 be an angle of size 45°. Let there be a rotation by 𝑑 degrees, where 𝑑< 0, about 𝑂. Find the images of the given figures. Answer the questions that follow.

a. What is the length of the rotated segment 𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛(𝐴𝐵)?

b. What is the degree of the rotated angle 𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛ሺ∠𝐶𝐷𝐸ሻ?

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7. Let 𝐿1 and 𝐿2 be parallel lines. Let there be a rotation by 𝑑 degrees, where −360 < 𝑑< 360, about 𝑂.

Is ሺ𝐿1ሻ′ ∥ ሺ𝐿2ሻ′?

8. Let 𝐿 be a line and 𝑂 be the center of rotation. Let there be a rotation by 𝑑 degrees, where 𝑑≠ 180 about 𝑂. Are the lines 𝐿 and 𝐿′ parallel?

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LESSON 5: ROTATIONS in the COORDINATE PLANE

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1) First, draw rectangle ABCD.A (2,7)B (8,7)C (8,2)D (2,2)

2) Next, rotate* (turn) ABCD 90 counter-clockwise about (around) the origin.

3) Then hold your paper at the origin and rotate your paper 90 counter-clockwise. Look at the NEW coordinates to find A’. Rotate your paper back to normal and plot A’. Repeat with B, C, and D.

4) Draw rectangle A’B’C’D’. 5) Last, label the coordinates of A’B’C’D’.

A’ (___,___)B’ (___,___)C ‘(___,___)D’ (___,___)

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Method 2: Use the RULES!!

Degree of Rotation ALWAYS

COUNTERCLOCKWISE

Pre-Image (Before)

Image (After)

90° ( x, y ) ( -y , x ) 180° ( x, y ) (-x, -y ) 270° ( x, y ) ( y , -x )

Draw quadrilateral QRST; Q(-8,7), R(-2,8), S(-2,2) and T(-8,0). Rotate QRST 180 about the origin. Use the rule (x, y) (-x , -y) Plot Q’R’S’T’ and draw rectangle Q’R’S’T’.

Q’ ( ___ , ___ )

R’ ( ___ , ___ )

S’ ( ___ , ___ )

T’ ( ___ , ___ )

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R

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1) First, draw triangle LMN.L ( 1, 4 )M ( 5, 4 )N ( 5, 1 )

2) Use the rules on the previous page to rotate (turn) triangle LMN 90 counter-clockwise about (around) the origin.

L’ (___,___)

M’ (___,___)

N ‘(___,___)

4) Draw triangle L’M’N’. 5) Label the coordinates of triangle L’M’N’

ROTATION SUMMARY

(1) A rotation maps a line to a line, a ray to a ray, a segment

to a segment, and an angle to an angle.

(2) A rotation preserves lengths of segments.

(3) A rotation preserves degrees of angles.

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