Rigid Transformations

Part 1: What Makes a Transformation "Rigid"?

Part 1 of 7 — Distance, Angle, and Orientation

Topics in This Part

🔑 Big Idea: A rigid transformation is not defined by what it looks like; it is defined by what it preserves. If two figures can be aligned by a rigid transformation, they are congruent — by definition.

What You'll Master in Part 1

• Decide whether a given coordinate rule is rigid without graphing it
• Predict which side lengths and angle measures change (and which don't)
• Distinguish translations, reflections, and rotations from dilations/stretches by their fingerprint on distance and orientation

🧭 The Three Invariants

A transformation $T$ is rigid (an isometry) if and only if it preserves every pairwise distance between points:

That single condition forces three observable consequences:

💡 What is not automatically preserved: orientation (clockwise vs. counter-clockwise ordering of vertices). Reflections reverse it; translations and rotations don't. We'll exploit this in the next part.

Why "preserves distance" is the right definition

You might be tempted to define rigidity as "preserves shape." But "shape" is vague — does a 90° rotation change the shape? It changes the picture you draw, but every measurement is identical. Distance preservation is the unambiguous mathematical condition, and everything else — angles, perimeter, area, congruence — falls out of it as a theorem, not a separate rule to memorize.

Check: Which Properties Survive a Rigid Transformation?

🔍 Rigid vs. Non-Rigid — A Coordinate-Rule Test

You can often decide whether a rule is rigid just by inspecting it.

Check: Reading Coordinate Rules

📐 Detecting Rigidity from a Mapping

Sometimes you are not given a formula — only a table that says "this point goes to that point." How do you tell whether such a mapping could extend to a rigid transformation?

The Test

Pick any two points $P, Q$ in the pre-image. Compute both

Check: Detecting Rigidity from a Table

🔄 Orientation: The Fingerprint That Tells Reflections Apart

The three rigid transformations of the plane split cleanly into two camps:

To measure orientation, list the vertices in order ($A, B, C$) and walk around the triangle. If you walk counter-clockwise, the orientation is positive; clockwise is negative.

Computing Orientation Algebraically

For triangle with vertices $A=(x_1,y_{}$, , , the signed quantity

Check: Diagnosing Transformations by Invariants

Part 1 Exit Quiz