Rotations - Geometry Lesson | Luna Learn 24

Rotations in Geometry

In this lesson, we will explore the concept of rotations in geometry. Rotations involve turning a shape around a fixed point by a specific angle. This transformation changes the position of the shape but maintains its size and shape.

Outline

Definition of Rotation
Rotation Angles
Examples
Exercises
Homework
Revision

1. Definition of Rotation

A rotation is a transformation that turns a shape around a fixed point, called the center of rotation. The amount of turning is specified by an angle, and the direction can be clockwise or counterclockwise.

2. Rotation Angles

Common angles of rotation include 90°, 180°, and 270°. Each angle has a specific effect on the shape:

90° clockwise: Rotates the shape to the right.
180°: Rotates the shape upside down.
270° clockwise: Rotates the shape to the left.

Example 1: Rotating a Point 90° Clockwise

Rotate the point \( (2, 3) \) 90° clockwise around the origin (0,0).

Using the formula for 90° clockwise rotation: \((x, y) \rightarrow (y, -x)\), \[ (2, 3) \rightarrow (3, -2) \] Thus, the new coordinates are \( (3, -2) \).

Example 2: Rotating a Point 180°

Rotate the point \( (-1, 4) \) 180° around the origin (0,0).

Using the formula for 180° rotation: \((x, y) \rightarrow (-x, -y)\), \[ (-1, 4) \rightarrow (1, -4) \] Thus, the new coordinates are \( (1, -4) \).

Example 3: Rotating a Shape 270° Counterclockwise

Rotate the triangle with vertices \( (1, 1) \), \( (1, 4) \), and \( (4, 1) \) 270° counterclockwise around the origin (0,0).

Using the formula for 270° counterclockwise rotation: \((x, y) \rightarrow (-y, x)\), \[ (1, 1) \rightarrow (-1, 1) \] \[ (1, 4) \rightarrow (-4, 1) \] \[ (4, 1) \rightarrow (-1, 4) \] Thus, the new vertices are \( (-1, 1) \), \( (-4, 1) \), and \( (-1, 4) \).

Example 4: Rotating a Point 90° Counterclockwise

Rotate the point \( (5, -3) \) 90° counterclockwise around the origin (0,0).

Using the formula for 90° counterclockwise rotation: \((x, y) \rightarrow (-y, x)\), \[ (5, -3) \rightarrow (3, 5) \] Thus, the new coordinates are \( (3, 5) \).

Example 5: Rotating a Rectangle 180°

Rotate the rectangle with vertices \( (2, 2) \), \( (2, 5) \), \( (5, 5) \), and \( (5, 2) \) 180° around the origin (0,0).

Using the formula for 180° rotation: \((x, y) \rightarrow (-x, -y)\), \[ (2, 2) \rightarrow (-2, -2) \] \[ (2, 5) \rightarrow (-2, -5) \] \[ (5, 5) \rightarrow (-5, -5) \] \[ (5, 2) \rightarrow (-5, -2) \] Thus, the new vertices are \( (-2, -2) \), \( (-2, -5) \), \( (-5, -5) \), and \( (-5, -2) \).

Example 6: Rotating a Hexagon 120°

Rotate the hexagon with vertices \( (1, 0) \), \( (2, 1) \), \( (2, 3) \), \( (1, 4) \), \( (0, 3) \), and \( (0, 1) \) 120° clockwise around the origin (0,0).

Using the formula for 120° clockwise rotation: \((x, y) \rightarrow (-\frac{1}{2}x - \frac{\sqrt{3}}{2}y, \frac{\sqrt{3}}{2}x - \frac{1}{2}y)\), \[ (1, 0) \rightarrow (-\frac{1}{2}, \frac{\sqrt{3}}{2}) \] \[ (2, 1) \rightarrow (-\frac{1 + \sqrt{3}}{2}, \frac{1 - \sqrt{3}}{2}) \] \[ (2, 3) \rightarrow (-\frac{5}{2}, \frac{3\sqrt{3} - 1}{2}) \] \[ (1, 4) \rightarrow (-\frac{7}{2}, \frac{7\sqrt{3} - 4}{2}) \] \[ (0, 3) \rightarrow (-\frac{3\sqrt{3}}{2}, \frac{3}{2}) \] \[ (0, 1) \rightarrow (-\frac{\sqrt{3}}{2}, \frac{1}{2}) \] Thus, the new vertices are approximately \( (-0.5, 0.866) \), \( (-1.366, -0.366) \), \( (-2.5, 1.732) \), \( (-3.5, 3.366) \), \( (-2.598, 1.5) \), and \( (-0.866, 0.5) \).

3. Exercises

Rotate the following shapes around the origin by the given degree:

Point \( (4, -1) \) 90° clockwise.
Triangle \( \triangle ABC \) with vertices \( A(2, 1) \), \( B(3, 4) \), and \( C(5, 2) \) 180° around the origin.
Square with vertices \( (0, 0) \), \( (0, 2) \), \( (2, 2) \), and \( (2, 0) \) 270° counterclockwise around the origin.

Solutions:

Point \( (4, -1) \) rotated 90° clockwise is \( (-1, -4) \).
Triangle \( \triangle ABC \) rotated 180° around the origin has vertices \( A(-2, -1) \), \( B(-3, -4) \), and \( C(-5, -2) \).
Square rotated 270° counterclockwise has vertices \( (2, 0) \), \( (2, -2) \), \( (0, -2) \), and \( (0, 0) \).

4. Homework

Complete the following exercises:

Rotate the point \( (3, 5) \) 90° counterclockwise around the origin.
Rotate the parallelogram with vertices \( (1, 2) \), \( (3, 5) \), \( (5, 3) \), and \( (3, 0) \) 180° around the origin.
Rotate the pentagon with vertices \( (2, 1) \), \( (3, 4) \), \( (5, 4) \), \( (6, 2) \), and \( (4, 0) \) 120° clockwise around the origin.

5. Revision

Review the key concepts of rotations:

Rotations turn a shape around a fixed point by a specific angle.
The direction of rotation can be clockwise or counterclockwise.
Common angles are 90°, 180°, and 270°.
The rotation does not change the size or shape of the figure, only its orientation.

Need more help? Contact us for online tutoring or text us to book a tutor. Don’t forget to follow us on Facebook and YouTube.

Back to Geometry Menu