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# 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

Step 3: Draw A’ so that PA’ = PA

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

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.
• 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.

### What we have learnt

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

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