Geometric Transformations and Composition
MathematicsVectors & Coordinate GeometryAges 16–17
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Vectors & Coordinate Geometry
Drag the F-shape and the elements of each transformation (translation vector, center of rotation, line of reflection, center of dilation) to see the image and its coordinate formula. Turn on composition mode to discover that two intersecting lines of reflection combine into a rotation through twice the angle between them, and two parallel lines combine into a translation — then check which transformations preserve distance and orientation.
Lesson: Geometric transformations in the plane
What it shows
An F-shape, which has no line of symmetry, is mapped by translation, reflection, point reflection, rotation and dilation, so any flip of the image is easy to spot. Translations, rotations and reflections are isometries: every length and angle is kept. Reflection is the only one that reverses orientation. A dilation with scale factor k multiplies distances by |k| and, for negative k, places the image on the opposite side of the center. Two reflections in intersecting lines combine into a rotation through twice the angle between them; parallel lines give a translation.
How to use
Choose a Transformation, then drag A, the center or the vector tip and compare the F with its image and the coordinate formula. For Rotation, change the Rotation angle / angle between the lines; for Dilation, move the Dilation scale factor k through 1, 0.5, −1 and −2. Finish with Composition of two reflections: drag d₂, read the resulting rotation angle, then tick Two parallel lines and measure the translation.
Parameters you can change
- Transformation Translation, Reflection, Point reflection, Rotation, Dilation, Composition of two reflections
- Rotation angle / angle between the lines -180–180 deg
- Dilation scale factor k -3–3
Questions to explore
- Which other transformation gives exactly the same image as a dilation with k = −1?
- Is the composition of two reflections ever itself a reflection? What does it become instead?
- Which transformations turn the F into its mirror image, and which only move or turn it?
