Introduction
Rotation is a transformation that turns a shape around a fixed point called the center of rotation. The amount of turn is measured in degrees and can be clockwise or counterclockwise (anti-clockwise).
In this lesson, we will learn how to rotate shapes in the coordinate plane about the origin (0,0) through angles of 90°, 180°, and 270°, both clockwise and counterclockwise. We will also learn how to determine the angle of rotation when given an object and its image.
Key Terms
Object: The original shape before rotation.
Image: The shape after rotation.
Center of rotation: The fixed point about which the rotation occurs.
Angle of rotation: The degree measure of the turn (90°, 180°, 270°).
Direction: Clockwise or counterclockwise.
Rotation Rules
Rotation Rules About the Origin
When rotating points about the origin (0,0), we can use these rules:
90° Counterclockwise (or 270° Clockwise)
\[ (x, y) \rightarrow (-y, x) \]Switch x and y coordinates, then change the sign of the new x-coordinate.
180° (Both Directions)
\[ (x, y) \rightarrow (-x, -y) \]Change the signs of both coordinates.
270° Counterclockwise (or 90° Clockwise)
\[ (x, y) \rightarrow (y, -x) \]Switch x and y coordinates, then change the sign of the new y-coordinate.
Remember
Clockwise rotation means turning in the same direction as clock hands move.
Counterclockwise (anti-clockwise) means turning in the opposite direction.
Positive angles typically indicate counterclockwise rotation.
Negative angles typically indicate clockwise rotation.
Drawing Rotations
Steps to Draw Rotations
- Plot the original shape (object) on the coordinate plane.
- Identify the center of rotation (usually the origin (0,0)).
- Apply the rotation rules to each vertex of the shape.
- Plot the new points (image) and connect them to form the rotated shape.
Example: 90° Counterclockwise Rotation
Rotate triangle ABC with vertices A(2,1), B(4,1), C(4,3) 90° counterclockwise about the origin.
Solution:
Using the rule (x,y) → (-y,x):
A(2,1) → A'(-1,2)
B(4,1) → B'(-1,4)
C(4,3) → C'(-3,4)
Plot both the original and rotated triangles.
Example: 180° Rotation
Rotate rectangle PQRS with vertices P(1,1), Q(3,1), R(3,2), S(1,2) 180° about the origin.
Solution:
Using the rule (x,y) → (-x,-y):
P(1,1) → P'(-1,-1)
Q(3,1) → Q'(-3,-1)
R(3,2) → R'(-3,-2)
S(1,2) → S'(-1,-2)
Plot both the original and rotated rectangles.
Practice Exercise
Question 1
Rotate triangle DEF with vertices D(1,3), E(3,3), F(2,5) 90° clockwise about the origin. State the coordinates of the image points.
Solution:
90° clockwise is equivalent to 270° counterclockwise. Using the rule (x,y) → (y,-x):
D(1,3) → D'(3,-1)
E(3,3) → E'(3,-3)
F(2,5) → F'(5,-2)
Question 2
Rotate quadrilateral GHIJ with vertices G(2,2), H(4,2), I(4,4), J(2,4) 270° counterclockwise about the origin. Draw both the object and image.
Solution:
Using the rule (x,y) → (y,-x):
G(2,2) → G'(2,-2)
H(4,2) → H'(2,-4)
I(4,4) → I'(4,-4)
J(2,4) → J'(4,-2)
Question 3
A triangle has vertices at K(1,1), L(1,4), M(3,1). Find the coordinates of its image after a 180° rotation about the origin.
Solution:
Using the rule (x,y) → (-x,-y):
K(1,1) → K'(-1,-1)
L(1,4) → L'(-1,-4)
M(3,1) → M'(-3,-1)
Ready for more challenges?
Full Practice SetDetermine Angle of Rotation
Steps to Determine Angle of Rotation
- Identify corresponding points on the object and image.
- Calculate the angle between lines connecting these points to the center of rotation.
- Determine if the rotation is clockwise or counterclockwise.
- Verify with other corresponding points to ensure consistency.
Example 1
Triangle ABC with vertices A(2,3), B(4,3), C(4,5) is rotated to A'(-3,2), B'(-3,4), C'(-5,4). Determine the angle and direction of rotation.
Solution:
Looking at point A(2,3) → A'(-3,2):
This matches the rule (x,y) → (-y,x), which is 90° counterclockwise.
Verifying with point B(4,3) → B'(-3,4) confirms the same rotation.
Example 2
Rectangle PQRS with vertices P(1,2), Q(3,2), R(3,4), S(1,4) is rotated to P'(-1,-2), Q'(-3,-2), R'(-3,-4), S'(-1,-4). Determine the rotation.
Solution:
Looking at point P(1,2) → P'(-1,-2):
This matches the rule (x,y) → (-x,-y), which is 180° rotation (direction doesn't matter for 180°).
All other points confirm this rotation.
Practice Exercise
Question 1
Point A(3,1) is rotated to A'(1,-3). Determine the angle and direction of rotation about the origin.
Solution:
A(3,1) → A'(1,-3) matches (x,y) → (y,-x), which is 270° counterclockwise or 90° clockwise.
Question 2
Triangle B(2,2), C(4,2), D(3,4) is rotated to B'(-2,2), C'(-2,4), D'(-4,3). Find the angle and direction of rotation.
Solution:
Looking at B(2,2) → B'(-2,2): This is reflection over y-axis, not a pure rotation.
Looking at C(4,2) → C'(-2,4): This doesn't match standard rotation rules.
This appears to be a combination of transformations, not a simple rotation.
Question 3
Point E(5,0) is rotated to E'(0,5). Determine the angle and direction of rotation about the origin.
Solution:
E(5,0) → E'(0,5) matches (x,y) → (-y,x), which is 90° counterclockwise.
Ready for more challenges?
Full Practice SetMath Challenge
Timed Test in 7 Levels
Test your math skills through progressively challenging levels