All Labs

Congruence & Transformations Lab

Investigate how rigid motions preserve distance and angle, test triangle congruence criteria, compose transformations, and detect polygon symmetry. Every computation happens in your browser with interactive coordinate-plane visualizations.

Guided Experiment: Rigid Motion Properties

Do rigid motions (translations, rotations, reflections) preserve distances and angle measures? Will the image always be congruent to the original?

Write your hypothesis in the Lab Report panel, then click Next.

Coordinate Plane

OriginalImage

Controls

dx3.0
dy2.0

Results

Translation (3, 2)
Distances preserved. Image is congruent to original.
Side Length Comparison
Side 1:3.0000→3.0000
Side 2:3.3541→3.3541
Side 3:3.3541→3.3541

Data Table

(0 rows)
#TrialOperationDistance PreservedOriginal SideImage SideCongruent
0 / 500
0 / 500
0 / 500

Reference Guide

Rigid Motions

A rigid motion (isometry) maps every point of a figure to a new location while preserving all distances and angle measures. The three rigid motions in the plane are translation, rotation, and reflection.

d(A,B)=d(A′,B′)d(A, B) = d(A', B')

Because distances are preserved, the image of any polygon under a rigid motion is congruent to the original.

Triangle Congruence Criteria

Two triangles are congruent when you can match all corresponding sides and angles. You do not need to check all six measurements. Any one of these shortcut criteria is sufficient.

  • SSS — three pairs of sides equal
  • SAS — two sides and the included angle equal
  • ASA — two angles and the included side equal
  • AAS — two angles and a non-included side equal
  • HL — hypotenuse and a leg of right triangles equal

Note that AAA (three matching angles) proves similarity but not congruence.

Composition of Transformations

Applying two or more rigid motions in sequence produces a composition. The composition of rigid motions is itself a rigid motion.

T2∘T1(P)=T2(T1(P))T_2 \circ T_1 (P) = T_2(T_1(P))

Key results include: two reflections over parallel lines equal a translation, and two reflections over intersecting lines equal a rotation by twice the angle between the lines.

Symmetry

A polygon has reflectional symmetry if a line maps it onto itself, and rotational symmetry of order n if rotating by 360/n degrees maps it onto itself.

A regular n-gon has n lines of symmetry and rotational symmetry of order n. A rectangle (non-square) has 2 lines of symmetry and rotational order 2.