Rotation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Rotation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A rigid transformation that turns every point of a figure by a fixed angle around a fixed center of rotation.

Like a Ferris wheel turning around its center hub—every seat traces a circle, staying the same distance from the axle while sweeping through the same angle.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Rotation preserves size and shape; changes position and orientation.

Common stuck point: Specify: center point, angle, and direction (clockwise or counter).

Sense of Study hint: Try placing your pencil tip on the center of rotation and spinning the paper. Note the angle and direction the shape moves.

Worked Examples

Example 1

easy
Rotate the point A(3, 2) by 90° counterclockwise about the origin. Where does A end up?

Solution

  1. 1
    Step 1: The rule for 90° CCW rotation is (x, y) \to (-y, x).
  2. 2
    Step 2: Apply to A(3, 2): A' = (-2, 3).
  3. 3
    Step 3: Verify distance from origin is unchanged: \sqrt{3^2+2^2} = \sqrt{13} and \sqrt{(-2)^2+3^2} = \sqrt{13}. ✓

Answer

A' = (-2, 3)
The 90° CCW rotation rule (x,y)\to(-y,x) is derived from the rotation matrix with \theta=90°: \cos90°=0, \sin90°=1, giving (-y, x). Rotation preserves distance from the center of rotation.

Example 2

medium
Rotate the point B(4, 0) by 60° counterclockwise about the origin. Give exact coordinates.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
What is the image of point (0, 5) after a 180° rotation about the origin?

Example 2

hard
Triangle with vertices A(1,0), B(3,0), C(2,2) is rotated 90° CCW about the point (1,0). Find the new vertices.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

transformation geoangles