Perform Rotations

Key Concepts

• Understand rotation
• Draw a rotation
• Apply coordinate rules for rotation
• Rotate a figure using coordinate rules

Rotation

Rotation is a transformation in which a figure is turned about a fixed point called the center of rotation. Rays drawn from the center of rotation to a point and their image form the angle of rotation. 

A rotation about a point A through an angle of x⁰ maps every point B in the plane to a point B’ so that one of the following properties is true. 

• If B is not the center of rotation A, then BP = B’P and m∠ BPB’= x⁰ or 
• If B is the center of rotation A, then the image of B is B. 

Direction of rotation

Rotations can be clockwise or counterclockwise. 

Note: 

In this chapter, all rotations are counterclockwise. 

Draw a rotation

Draw a 120⁰ rotation of ΔABC about P. 

Solution: 

Step 1: Draw a segment from A to P. 

Step 2: Draw a ray to form a 120⁰ PA

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Step 3: Draw A’ so that PA’ = PA 

Step 4: Repeat steps 1-3 for each vertex. Draw ΔA′B′C’

Rotations about the origin

If a rotation is shown in a coordinate plane, the center of rotation is the origin. 

The diagram shows rotations of point A 130°, 220°, and 310° about the origin. A rotation of 360° returns a figure to its original coordinates. 

There are coordinate rules that can be used to find the coordinates of a point after rotations of 90°, 180°, or 270° about the origin. 

Coordinate rules for rotations about the origin: 

When a point (a, b) is rotated counterclockwise about the origin, the following are true: 

1. For a rotation of 90°°, (a, b) → (–b, a). 
2. For a rotation of 180°°, (a, b) → (–a, –b). 
3. For a rotation of 270°°, (a, b) → (b, –a). 

Example: 

When a point (3, 4) is rotated counterclockwise about the origin. Find the coordinate after the rotation of 90°, 180° rotation. 

Solution: 

For a rotation of 90°, (a, b) → (–b, a). 

(3,4) → (–4, 3) 

For a rotation of 180°, (a, b) → (–a, –b). 

(3,4) → (–3, –4) 

For a rotation of 270°, (a, b) → (b, –a). 

(3,4) → (4, –3) 

Rotate a figure using the coordinate rules: 

Now, let us use the coordinate rules to rotate a figure in the coordinate plane. 

Example 1: 

Graph quadrilateral ABCD with vertices A(3, 1), B(5, 1), C(5, –3), and D(2, –1). Then rotate the quadrilateral 270° about the origin. 

Solution: 

Graph ABCD. Use the coordinate rule for a 270° rotation to find the images of the vertices. 

(a, b) → (b, –a) 

A(3, 1) → A’(1, –3) 

B(5, 1) → B’(1, –5) 

C(5, –3) → C’(–3, –5) 

D(2, –1) → D’(–1, –2) 

Now graph the image A’B’C’D’. 

Example 2: 

Graph a triangle ABC with vertices A(3, 0), B(4, 3), and C(6, 0). Rotate the triangle 90° about the origin. 

Solution: 

Graph ABC. Use the coordinate rule for a 90° rotation to find the images of the vertices. 

(a, b) → (-b, a) 

A(3, 0) → A’(0,3) 

B(4, 3) → B’(–3, 4) 

C(6, 0) → C’(0, 6) 

Now graph the image A’B’C’. 

Example 3: 

Graph quadrilateral ABCD with vertices A(3, 1), B(5, 1), C(5, –3), and D(2, –1). Then rotate the quadrilateral 180° about the origin. 

Solution: 

Graph ABCD. Use the coordinate rule for a 180° rotation to find the images of the vertices. 

(a, b) → (–a, –b) 

A(3, 1) → A’(–3, –1) 

B(5, 1) → B’(–5, -1) 

C(5, –3) → C’(–5, 3) 

D(2, –1) → D’(–2, 1) 

Now graph the image A’B’C’D’. 

Summary

• Rotation is a transformation in which a figure is turned about a fixed point called the center of rotation. Rays drawn from the center of rotation to a point and their image form the angle of rotation.
• Rotations can be clockwise or counterclockwise.
• If a rotation is shown in a coordinate plane, the center of rotation is the origin.
• When a point (a, b) is rotated counterclockwise about the origin, the following are true:
  • For a rotation of 90°,(a, b) + (-b, a).
  • For a rotation of 180°,(a, b) + (-a, -b)
  • For a rotation of 270°,(a, b) → (b,-a).

Exercise

• Draw a 90° rotation of AABC about A. AB= 3 cm, BC= 4 cm and AC= 5 cm.
• Draw a 180° rotation of AABC about A. AB= 3 cm, BC=4 cm and AC=5 cm
• When a point (8,-3) is rotated counterclockwise about the origin. Find the coordinate after the rotation of 90°, 1800,006) rotation.
• When a point (-1,-4) is rotated counterclockwise about the origin. Find the coordinate after the rotation of 90°, 180°,006 rotation.
• Trace ADEF and P. Then draw a 50° rotation of ADEF about P.

• Graph ARST with vertices R(3,0), S(4,3), and T6,0). Rotate the triangle 90° about the origin.
• Graph ARST with vertices R(3,0), S(4,3), and T6,0). Rotate the triangle 270° about the origin.
• Rotate the figure the given number of degrees about the origin. List the coordinates of the vertices of the image.

• Rotate the figure the given number of degrees about the origin. List the coordinates of the vertices of the image.

• Rotate the figure the given number of degrees about the origin. List the coordinates of the vertices of the image.

Concept Map

What we have learnt

• Understand rotation
• Draw a rotation
• Apply coordinate rules for rotation
• Rotate a figure using coordinate rules.

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