ROTATION

Concept: A single turn of a line or figure by a given angle around an axis or a fixed point.

Steps for Solving Rotation Problems

Step 1 | Plot the given points on a graph.
Step 2 | Connect the plotted points, forming the described figure.
Step 3 | Begin at the fixed point. Pick one line in the figure that includes the fixed point. Determine the length of that line.
Step 4 | Rotate that line the designated number of degrees, using the fixed point as your axis.
Step 5 | Ensure that the rotated line is equal in length to the original line, and note the coordinates of the end-point of the rotated line.
Step 6 | Repeat Steps 3, 4, and 5, with the other line that includes the fixed point.
Step 7 | If the figure is a triangle (has three coordinates), connect the endpoints of the rotated lines, and you have drawn your rotated triangle. Proceed to Step 11.
Step 8 | If the figure has four coordinates, drawn a dotted line between the fixed point and the fourth coordinate. Repeat Steps 3, 4, and 5 with the dotted line.
Step 9 | Connect the endpoint of the first line rotated to the new fourth coordinate.
Step 10 | Connect the new fourth coordinate to the endpoint of the second line rotated.
Step 11 | List the new coordinates of the rotated figure.

Example 1 A triangle with coordinates A(4, 10), B(4, 6), and C(10, 6) is rotated clockwise by while keeping point B fixed. What are the new coordinates of the triangle?

Step 1 | Plot the given points on a graph. | A, B, and C
Step 2 | Connect the plotted points, forming the described figure. | ABC
Step 3 | Begin at the fixed point. | B is the fixed point.

Step 3 (cont.) | Pick one line in the figure which includes the fixed point. |
Step 3 (cont.) | Determine the length of that line. Count upwards from point B until point A is reached. | 4

Step 4 | Rotate that line the designated number of degrees, using the fixed point ( B )as your axis. | Rotate clockwise

Step 5 | Ensure that the rotated line is equal in length to the original line, and note the coordinates of the end-point of the rotated line. | (0, 6)

Step 6 | Repeat Steps 3, 4, and 5, with the other line that includes the fixed point. |
Step 6 Repeated Step 3 | Begin at the fixed point. | B is the fixed point.
Step 6 Repeated Step 3 continued | Pick one line in the figure which includes the fixed point. |
Step 6 Repeated Step 3 continued | Determine the length of that line. |
Step 6 Repeated Step 3 continued | Use the Pythagorean Theorem to determine the distance from point B to point C. a2 + b2 = c2 42 + 22 = c2 and 16 + 4 = c2 20 = c2 4.48 = c |
4.48 = c is rounded to 4.5 = c

Step 6 Repeated Step 4 | Rotate that line the designated number of degrees, using the fixed point ( B )as your axis. | Rotate clockwise

Step 6 Repeated Step 5 | Ensure that the rotated line is equal in length to the original line, and note the coordinates of the end-point of the rotated line. | (0, 8)

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

10

B

A

C

Step 7

C

A

Step 7 | If the figure is a triangle (has three coordinates), connect the endpoints of the rotated lines, and you have drawn your rotated triangle. Proceed to Step 11. | Connect points A and C to form
Step 8 | If the figure has four coordinates, drawn a dotted line between the fixed point and the fourth coordinate. Repeat Steps 3, 4, and 5 with the dotted line. | Does not apply.
Step 9 | Connect the endpoint of the first line rotated to the new fourth coordinate. | Does not apply.
Step 10 | Connect the new fourth coordinate to the endpoint of the second line rotated. | Does not apply.
Step 11 | List the new coordinates of the rotated figure. | A = (0, 6) B = (4, 6) C = (0, 8)

Example 2 A rectangle with coordinates A(5, 7), B(9, 7), C(5, 4) and D(9, 4) is rotated counterclockwise by , while keeping Point A fixed. What are the new coordinates of the rectangle?

Step 1 | Plot the given points on a graph. | A, B, C, and D
Step 2 | Connect the plotted points, forming the described figure. | ABCD
Step 3 | Begin at the fixed point. | A is the fixed point.

Step 3 continued | Pick one line in the figure which includes the fixed point. | = picked line
Step 3 continued | Determine the length of that line. Count from point A to point B. | 4

Step 4 | Rotate that line the designated number of degrees, using the fixed point as your axis. | Rotate counterclockwise

Step 5 | Ensure that the rotated line is equal in length to the original line, and note the coordinates of the end-point of the rotated line. | (5, 3)

Step 6 | Repeat Steps 3, 4, and 5, with the other line that includes the fixed point. |
Step 6 Repeated Step 3 | Begin at the fixed point. | A is the fixed point
Step 6 Repeated Step 3 continued | Pick one line in the figure which includes the fixed point. |
Step 6 Repeated Step 3 continued | Determine the length of that line. | 3

Step 6 Repeated Step 4 | Rotate that line the designated number of degrees, using the fixed point ( A )as your axis. | Rotate counterclockwise

Step 6 Repeated Step 5 | Ensure that the rotated line is equal in length to the original line, and note the coordinates of the end-point of the rotated line. | (2, 7)

Step 7 | If the figure is a triangle (has three coordinates), connect the endpoints of the rotated lines, and you have drawn your rotated triangle. Proceed to Step 11. | Does not apply.
Step 8 | If the figure has four coordinates, draw a dotted line between the fixed point and the fourth coordinate ( D). Repeat Steps 3, 4, and 5 with the dotted line. | dotted line between the fixed point and the fourth coordinate

Step 8 Repeated Step 3 | Begin at the fixed point. | A is the fixed point
Step 8 Repeated Step 3 continued | Pick one line in the figure which includes the fixed point. | line AD
Step 8 Repeated Step 3 continued | Determine the length of that line. Use the Pythagorean Theorem to determine the distance from point A to point D. a2 + b2 = c2 42 + 32 = c2 and 16 + 9 = c2 25 = c2 5 = c |

Step 8 Repeated Step 4 | Rotate that line the designated number of degrees, using the fixed point ( A )as your axis. | Rotate line AD counterclockwise

Step 8 Repeated Step 5 | Ensure that the rotated line is equal in length to the original line, and note the coordinates of the end-point of the rotated line. {Use the Pythagorean Theorem to determine length, as in Step 8, repeated Step 3 continued.} | (2, 3)

Step 9 | Connect the endpoint of the first line rotated to the new fourth coordinate. (Connect point C to point D and form line CD.) | line CD
Step 10 | Connect the new fourth coordinate to the endpoint of the second line rotated. (Connect point D to point B and form line DB.) | line DB
Step 11 | List the new coordinates of the rotated figure. | A (5, 7) B (5, 3) C (2, 7) D (2, 3)