Step 1: Make the center the new origin

Subtract the center from the point.

(Point) - (Center)

This tells you where the point is relative to the center.

Step 2: Rotate

Apply the rotation rule to the new coordinates.

90° CCW:

(x,y) → (-y,x)

90° CW:

(x,y) → (y,-x)

180°:

(x,y) → (-x,-y)

Step 3: Put it back

Add the center back.

(Rotated Point) + (Center)

Problem 1 (Origin)

Rotate:

(4,1)

90° CCW about the origin.

Step 1

Center:

(0,0)

Subtract center:

(4,1)

(no change)

Step 2

Apply 90° CCW:

(4,1)

↓

(-1,4)

Step 3

Add center back:

(-1,4)+(0,0)

=

(-1,4)

Answer

(-1,4)

Problem 2

Rotate:

(8,5)

90° CCW about:

(5,4)

Step 1

Make (5,4) the new origin.

(8,5)-(5,4)

=

(3,1)

Step 2

Rotate:

(3,1)

↓

(-1,3)

Step 3

Add center back:

(-1,3)+(5,4)

=

(4,7)

Answer

(4,7)

Problem 3

Rotate:

(7,2)

90° CW about:

(4,1)

Step 1

Subtract center:

(7,2)-(4,1)

=

(3,1)

Step 2

90° CW:

(3,1)

↓

(1,-3)

Step 3

Add center back:

(1,-3)+(4,1)

=

(5,-2)

Answer

(5,-2)

Problem 4

Rotate:

(10,6)

180° about:

(7,4)

Step 1

Subtract center:

(10,6)-(7,4)

=

(3,2)

Step 2

180°:

(3,2)

↓

(-3,-2)

Step 3

Add center back:

(-3,-2)+(7,4)

=

(4,2)

Answer

(4,2)

Problem 5

Rotate:

(1,7)

90° CCW about:

(3,4)

Step 1

Subtract center:

(1,7)-(3,4)

=

(-2,3)

Interpretation:

The point is:

2 left
3 up

from the center.

Step 2

Rotate 90° CCW:

(-2,3)

↓

(-3,-2)

Step 3

Add center back:

(-3,-2)+(3,4)

=

(0,2)

Answer

(0,2)

Summary Sheet for Students

Rotations Around Any Point

1. Make the center the new origin

Subtract the center.

Point - Center

2. Rotate

90° CCW

(x,y) → (-y,x)

90° CW

(x,y) → (y,-x)

180°

(x,y) → (-x,-y)

3. Put the point back

Add the center.

Rotated Point + Center