Lesson 4.1.7

Rotations About a Non-Origin Point

Not every rotation spins around the origin. When the center of rotation is at another point , you use a three-step translate–rotate–translate strategy to leverage the rules you already know.

Introduction

In Lesson 4.1.6 you rotated about the origin because the rules were simple. Real-world rotations, however, rarely pivot at . A door rotates around its hinges; a Ferris wheel rotates around its axle. This lesson teaches the general method that works for any center of rotation.

Past Knowledge

Origin rotations (4.1.6). Translations (4.1.2–4.1.3).

Today's Goal

Rotate points and figures about any center using the translate–rotate–translate method.

Future Success

Compositions of transformations (4.2.2), rotational symmetry (4.2.1), and congruence proofs.

Key Concepts

The Three-Step Method

To rotate a point by angle about center :

  1. Translate the center to the origin:
  2. Rotate using the standard origin rule for angle
  3. Translate back: add to the result

Formulae for Common Angles

Let the center be and the point be . Define and .

Angle (CCW)Image Point
90°
180°
270°

Why It Works

Translating the center to the origin temporarily repositions the figure so we can use the simple origin-rotation rules. Translating back puts everything in its correct final location. The rotation itself is unchanged because translations don't alter angles or distances.

Worked Examples

Basic

90° CCW About a Given Center

Rotate by 90° CCW about center .

Step 1 — Translate:

Step 2 — Rotate 90° CCW: using

Step 3 — Translate back:

Rotateabout(,)
P(5,2)→P′(2,7)
Intermediate

180° Rotation of a Segment

Rotate with and by 180° about .

Point A:

Translate:

Rotate 180°:

Translate back:

Point B:

Translate:

Rotate 180°:

Translate back:

Rotateabout(,)
A(3,1)→A′(5,5)
B(7,5)→B′(1,1)
Advanced

270° CCW (= 90° CW) About a Center

Rotate by 270° CCW about .

Step 1 — Translate:

Step 2 — Rotate 270° CCW:

Step 3 — Translate back:

Rotateabout(,)
Q(-2,4)→Q′(4,4)

Common Pitfalls

Forgetting to Translate Back

The most common error! After rotating about the origin (Step 2), you must add back. Without Step 3, your image is displaced from where it should be.

Translating in the Wrong Direction

Step 1 subtracts ; Step 3 adds . Mixing these up (adding in Step 1, subtracting in Step 3) doubles the error.

Using Origin Rules Directly

The rules etc. only work when the center is the origin. If you apply them without translating first, you'll get the wrong answer every time.

Real-Life Applications

Ferris Wheels

Each seat on a Ferris wheel follows a rotation about the wheel's axle — which is definitely not the origin. Engineers use the translate–rotate–translate method to compute the exact position of every seat at any moment.

Hinged Doors & Swing Arms

A door rotates about its hinges. A robot arm rotates about its shoulder joint. In each case, the pivot point is not the origin, so the three-step translate–rotate–translate strategy is exactly the method used by engineers to compute positions.

Practice Quiz

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