Transformation playground
Translate, rotate, reflect, or scale a triangle and compare its coordinates and area.
Your measurements
The original triangle is A(−2, −1), B(2, −1), C(0, 2). Rotations and dilations are centered at the origin.
Why it works
Transformation rules on the coordinate plane
Every rule below acts on the original point P(3, 1). Rotations turn about the origin; positive angles turn counterclockwise. Translations, reflections, and rotations are rigid transformations (isometries): they keep lengths, angles, and area. A dilation keeps angles but scales lengths.
| Transformation | Coordinate rule | P(3, 1) → |
|---|---|---|
| TranslationTranslate by (a, b) (a = −4, b = 2) | (x, y) → (x + a, y + b) | (−1, 3) |
| ReflectionReflect over the x-axis | (x, y) → (x, −y) | (3, −1) |
| ReflectionReflect over the y-axis | (x, y) → (−x, y) | (−3, 1) |
| ReflectionReflect over y = x | (x, y) → (y, x) | (1, 3) |
| ReflectionReflect over y = −x | (x, y) → (−y, −x) | (−1, −3) |
| ReflectionReflect over the line x = h (h = 2) | (x, y) → (2h − x, y) | (1, 1) |
| ReflectionReflect over the line y = k (k = 2) | (x, y) → (x, 2k − y) | (3, 3) |
| RotationRotate 90° counterclockwise | (x, y) → (−y, x) | (−1, 3) |
| RotationRotate 180° | (x, y) → (−x, −y) | (−3, −1) |
| RotationRotate 270° counterclockwise (90° clockwise) | (x, y) → (y, −x) | (1, −3) |
| DilationDilate by k from the origin (k = 2) | (x, y) → (kx, ky) | (6, 2) |
Transformation playground: common questions
Wondering about the why? Start here.
Which transformations keep a triangle the same size?
Translations, rotations, and reflections preserve lengths, angles, and area. A dilation changes lengths by its scale factor while preserving angles. In this playground, rotations and dilations are centered at the origin.
Learn the transformation rulesWhy does doubling a shape’s size quadruple its area?
A dilation by a positive factor k multiplies each length by k, so area is multiplied by k². Doubling every length makes the area four times as large; halving every length makes it one quarter as large.
Explore similarity and scale