In the Introduction to High School Geometry in the CCSSM, we read this:

The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation. Fundamental are the rigid motions: translations, rotations, reflections, and combinations of these, all of which are here assumed to preserve distance and angles (and therefore shapes generally).

This excerpt raises several questions: what exactly is a transformation? What is a rigid motion? And what does it mean to preserve distance and angles? Let’s take these questions one at a time.

What is a transformation of the plane? Standard G-CO.2 speaks directly to this question: we read that students must be able to “describe transformations as functions that take points in the plane as inputs and give other points as outputs.” This same standard also calls for students to be able to “compare transformations that preserve distance and angle to those that do not.” In other words, students should be able to determine which transformations are rigid motions and which are not. Let’s give this term a precise definition.

Definition: a rigid motion is a transformation of the plane that preserves distance and angles.

What does it mean to say that a transformation preserves distance?

Let T be a transformation, and let A and B be any two points in the plane. T is a distance-preserving function (or isometry) if the distance between T(A) and T(B) is the same as the distance between A and B. That is, A and B remain the same distance apart even after being transformed.

Which of the familiar transformations are rigid motions?

Axiom 1: reflections are rigid motions.

reflection preserves distance

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Axiom 2: rotations are rigid motions.

rotations preserve distance

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Axiom 3: translations are rigid motions.

translations preserve distance

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What does it mean to say that a transformation preserves angles?

Let T be a transformation, and let \alpha be any angle in the plane. T preserves angles if the measure of T(\alpha) is the same as the measure of \alpha. That is, the measure of \alpha remains the same even after being transformed.

reflections preserve angles

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There you have it! The facts outlined above will be used repeatedly as we develop theorems about congruent figures. Together with the definitions of reflection, rotation, and translation, the axioms above form the backbone of the theory of congruence by rigid motions.

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