🎯⭐ INTERACTIVE LESSON

Rigid Transformations

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Rigid Transformations - Complete Interactive Lesson

Part 1: What Makes a Transformation Rigid?

Part 1: What Makes a Transformation "Rigid"?

Part 1 of 7 — Distance, Angle, and Orientation

Topics in This Part

Section
The Three Invariants
Rigid vs. Non-Rigid
Reading a Coordinate Rule
Detecting Rigidity from a Mapping
Orientation: Direct vs. Opposite

🔑 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:

|T(P) - T(Q)| \;=\; |P - Q| \quad \text{for all points } P, Q.

That single condition forces three observable consequences:

What is preserved	Why it follows from distance preservation
Segment lengths	A segment is determined by its two endpoints; distance is preserved.
Angle measures	An angle is fixed by the three distances between its three defining points (Law of Cosines).
Shape & area	Lengths and angles together determine the figure up to position.

💡 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.

Rule	Rigid?	Quick reason
$(x,y) \to (x+3,\, y-2)$	✅ Yes	Translation — adds the same vector to every point.
$(x,y) \to (-x,\, y)$	✅ Yes	Reflection over the y-axis.
$(x,y) \to (-y,\, x)$	✅ Yes	90° rotation about the origin.
$(x,y) \to (2x,\, 2y)$	❌ No	Dilation — doubles every distance from the origin.
$(x,y) \to (2x,\, y)$	❌ No	Horizontal stretch — stretches some distances but not others.
$(x,y) \to (x+y,\, y)$	❌ No	Shear — preserves area but not distance.

The Linear-Algebra Fingerprint (Optional Insight)

Every transformation of the plane that fixes the origin can be written as

\begin{pmatrix} x' \\ y' \end{pmatrix} \;=\; M \begin{pmatrix} x \\ y \end{pmatrix}

for some $2 \times 2$ matrix $M$ . The transformation is rigid (fixing the origin) iff $M$ is orthogonal: $M^\top M = I$ . Translations are then layered on top by adding a vector. You don't need this for the AP-style work in this unit, but it explains why there are exactly three families of rigid transformations (translation, reflection, rotation) — and no others.

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

d_{\text{before}} = |P - Q| \quad \text{and} \quad d_{\text{after}} = |P' - Q'|.

If $d_{\text{before}} \neq d_{\text{after}}$ for any pair, the mapping is not rigid. (One bad pair is enough to disqualify it.)

If all pairwise distances match, the mapping extends to exactly one rigid transformation of the plane.

Worked Example

Point	Image
$A = (0,0)$	$A' = (3,1)$
$B = (4,0)$	$B' = (3,5)$
$C = (4,3)$	$C' = (0,5)$

Check the three pairwise distances:

Pair	Before	After
$AB$	$4$	$\sqrt{0^2 + 4^2} = 4$ ✅
$BC$	$3$	$\sqrt{3^2 + 0^2} = 3$ ✅
$AC$	$5$	$\sqrt{3^2 + 4^2} = 5$ ✅

All three match → the mapping is consistent with a rigid transformation. (In fact it's a 90° rotation about $(2, -1)$ — but we don't need to identify which rigid transformation to conclude that one exists.)

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:

Family	Orientation	Example
Direct (orientation-preserving)	Vertex order unchanged	Translation, rotation
Opposite (orientation-reversing)	Vertex order flipped	Reflection

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 $A=(x_1,y_1)$ , $B=(x_2,y_2)$ , $C=(x_3,y_3)$ , the signed quantity

2 \cdot \text{Area}_{\text{signed}} \;=\; (x_2 - x_1)(y_3 - y_1) - (x_3 - x_1)(y_2 - y_1)

is positive for counter-clockwise vertices and negative for clockwise vertices.

A translation or rotation preserves the sign.
A reflection flips the sign.
A dilation by positive scale factor preserves the sign; by negative scale factor flips it.

🔑 Use this to identify mystery transformations. If a transformation preserves all distances and preserves orientation, it's a translation or rotation. If it preserves all distances but reverses orientation, it must contain at least one reflection.

Check: Diagnosing Transformations by Invariants

Part 1 Exit Quiz

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