## 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.

.

#### Related topics

#### Composite Figures – Area and Volume

A composite figure is made up of simple geometric shapes. It is a 2-dimensional figure of basic two-dimensional shapes such as squares, triangles, rectangles, circles, etc. There are various shapes whose areas are different from one another. Everything has an area they occupy, from the laptop to your book. To understand the dynamics of composite […]

Read More >>#### Special Right Triangles: Types, Formulas, with Solved Examples.

Learn all about special right triangles- their types, formulas, and examples explained in detail for a better understanding. What are the shortcut ratios for the side lengths of special right triangles 30 60 90 and 45 45 90? How are these ratios related to the Pythagorean theorem? Right Angle Triangles A triangle with a ninety-degree […]

Read More >>#### Ways to Simplify Algebraic Expressions

Simplify algebraic expressions in Mathematics is a collection of various numeric expressions that multiple philosophers and historians have brought down. Talking of algebra, this branch of mathematics deals with the oldest concepts of mathematical sciences, geometry, and number theory. It is one of the earliest branches in the history of mathematics. The study of mathematical […]

Read More >>#### How to Solve Right Triangles?

In this article, we’ll learn about how to Solve Right Triangles. But first, learn about the Triangles. Triangles are made up of three line segments. These three segments meet to form three angles. The lengths of the sides and sizes of the angles are related to one another. If you know the size (length) of […]

Read More >>
Comments: