## 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 x^{0} 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’= x
^{0}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 120^{0} rotation of ΔABC about P.

**Solution: **

**Step 1**: Draw a segment from A to P.

**Step 2: **Draw a ray to form a 120^{0} PA

.

**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:

- 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).

**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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