In these lessons, we will learn

- what is rotation
- how to draw the rotated image of an object given the center, the angle and the direction of rotation.
- how to find the angle of rotation given the object, its image and the center of rotation.
- how to rotate points and shapes on the coordinate plane about the origin.

**Related Pages**

Transformations In Math

Geometric Transformations

Transformations In The Coordinate Plane

Translation

More Geometry Lessons

A **rotation** is a transformation in which the object is rotated
about a fixed point.

The **direction of rotation** can be **clockwise**
or **anticlockwise**.

The fixed point in which the rotation takes place is called the **center of rotation**.
The amount of rotation made is called the **angle of rotation**.

**Example:**

For any rotation, we need to specify the center, the angle and the direction of rotation.

Given the center of rotation and the angle of rotation we can determine the rotated image of an object.

**Example:**

Determine the image of the straight line XY under an anticlockwise rotation of 90˚ about O.

**Solution:**

Step 1: Join point X to O.

Step 2: Using a protractor, draw a line 90˚ anticlockwise from the line OX. Mark on the line the point X’ such that the line of OX’ = OX

Step 3: Repeat steps 1 and 2 for point Y. Join the points X’ and Y’ to form the line X’Y’.

Given an object, its image and the center of rotation, we can find the angle of rotation using the following steps.

Step 1: Choose any point in the given figure and join the chosen point to the center of rotation.

Step 2: Find the image of the chosen point and join it to the center of rotation.

Step 3: Measure the angle between the two lines. The sign of the angle depends on the direction of rotation. Anti-clockwise rotation is positive and clockwise rotation is negative.

**Example:**

Figure A’B’C’ is the image of figure ABC. O is the center of rotation.
Find the angle of rotation.

**Solution:**

Step 1: Join A to O.

Step 2: Join A’ to O.

Step 3: Measure the angle AOA’.

The angle of rotation is 62˚ anticlockwise or +62˚

**Manual rotation of a polygon about a given point at a given angle**

You will need a straightedge, a protractor, and a compass. We will perform rotations about a point inside the figure, one outside the figure and one on the figure.

**Examples:**

- Rotate ABCD by 143° about point P.
- Rotate JKLM by 111° about point Q.

**How to rotate a figure around a fixed point using a compass and protractor?**

You are given:

- A center of rotation.
- A figure to rotate.
- An angle of rotation.

- Draw a ray from the center of rotation to the point you wish to rotate.
- Draw an angle with the center of rotation as the vertex.
- Use a compass to draw a circle (arc) with the center at the center of rotation and a radius from the center of rotation to the point you are rotating.
- Now rotate all the other points and connect the dots.

We will now look at how points and shapes are rotated on the coordinate plane. It will be helpful to note the patterns of the coordinates when the points are rotated about the origin at different angles.

A rotation is an isometric transformation: the original figure and the image are congruent. The orientation of the image also stays the same, unlike reflections. To perform a geometry rotation, we first need to know the point of rotation, the angle of rotation, and a direction (either clockwise or counterclockwise). A rotation is also the same as a composition of reflections over intersecting lines.

The following diagrams show rotation of 90°, 180° and 270° about the origin. Scroll down the page for more examples and solutions.

**How to rotate points on the coordinate plane?**

The following videos show clockwise and anticlockwise rotation of 0˚, 90˚, 180˚ and 270˚about the origin (0, 0). The pattern of the coordinates are also explored.

**Reflection in intersecting lines Theorem**

If lines k and m intersect at a point P, then a reflection in k followed by a reflection in m is
the same as a rotation about point P.

The angle of rotation is 2x° where x° is the measure of the acute angle formed by the lines k and m.

Two reflections in parallel lines = translation.

Two reflections in intersecting lines = rotation.

**Example:**

Below is a composition of two reflection in intersecting lines. What is a single transformation that
would map ABCD onto A"B"C"D"?

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