∑ Mathematics LabTransformations · v4.0.4

MATHEMATICS LAB · GEOMETRY

Geometric Transformations in the Plane

Move the vertices, change a transformation, and compare the original and transformed figures. Explore vectors, centers, axes, and compositions with mathematical steps.

Coordinates are real numbers and angles are measured in degrees. Positive rotations are counterclockwise; compositions apply T₁ first, then T₂.

1. Define the figure and transformation

Drag the vertices or edit the coordinates

Initial figure

Point A
Point B
Point C

First transformation T₁

A translation moves each vertex by the same vector.
Examples:

Display options

Controls and information

Drag A, B, C (and D) to change the figure; drag the center or the two points defining theaxis to change the transformation. Use +, −, and Fit to zoom. For a translation, drag the vector tip.

Initial figure Final figure Intermediate figure Center or axis Correspondences

3. Composition of transformations

T₁ first, then T₂

2. Interactive graph

Axes at 1:1 scale
Original and transformed figures with constructions
Interactive geometric transformationInitial and transformed polygons on the Cartesian plane. Drag vertices or geometric parameters.
100%
Transformation completed

Transformed figure coordinates

The figure gradually follows the selected transformation.

4. Results

Coordinates and properties

5. Geometric construction

Meaning of the transformation

6. Explained calculations

Matrices and numerical substitutions

What you can study in this version

Isometries

Translations, rotations, and reflections, preserving lengths and areas.

Homotheties

Enlargements, reductions, and negative ratios, with scale factors and area changes.

Compositions

Two successive transformations, an intermediate figure, and affine matrix multiplication in the correct order.

The lab handles triangles and quadrilaterals, including irregular ones. Angles and decimals use numerical approximations; the formulas explain each step rather than replace a formal proof.