A rotation turns a figure about a fixed point. This page works through rotations in the coordinate plane: the direction conventions for positive and negative angle measures, the standard 90°, 180°, and 270° rotations about the origin, rotations about any point, and rotational symmetry — when a figure can be rotated by 180° or less and map directly onto itself.
What you are looking at: a triangle ABC on a coordinate grid, with the origin marked as the center of rotation. The original triangle stays on the grid as a light outline, and the rotated image is drawn in blue.
What to do: press a rotation button. The triangle is animated through the turn — watch which way it travels. Each new press turns the figure from where it sits now, and the table under the grid shows the vertices before and after the most recent turn. Press Reset to return the figure to its starting position.
What to notice: the direction of travel matters, not just the ending place. A turn of 90° clockwise travels below the starting position on its way around, while a turn of 90° counterclockwise travels above it — even though other turns can land a figure in the same final place.
What you are looking at: a triangle ABC and a center of rotation P that is not the origin. The coordinate rules for rotations are written for the origin, so a rotation about any other point is done in three steps, the method from the lesson notes.
What to do: work through the three steps. Step 1: translate every vertex so the center lands at the origin. Step 2: apply the origin rotation rule to the translated vertices. Step 3: translate back by the opposite amount to get the final image. Choose the correct set of vertices at each step. A first wrong choice earns a hint and a retry; after a second miss the correct set is shown and the activity moves on.
What to notice: the center itself never moves — after Step 1 it sits at the origin, and after Step 3 it is back where it started. Every other point keeps its distance from the center through all three steps.
What you are looking at: a shape drawn in blue, with its original outline shown as a dashed grey outline underneath. The shape turns about its own center, which is marked with a dot.
What to do: choose a shape, then press Rotate to turn it one step at a time. Each time the blue shape lands exactly on its dashed outline, that angle is recorded under the grid. One full turn brings every shape back to its start, so the interesting question is what happens at 180° or less. When you are ready, answer the question under the grid: the smallest angle, 180° or less, at which the shape maps directly onto itself.
What to notice: landing back at the start after a full 360° turn does not count — every shape does that. Rotational symmetry is about the smaller turns in between.
What you are looking at: a blue triangle, a dashed red target outline, and a center of rotation. The task above the grid says what kind of challenge it is.
What to do: for a rotation about the origin, drag the orange handle to turn the blue triangle. The triangle turns with your pointer, and when you release, it snaps to the nearest exact quarter-turn — 90°, 180°, or 270°. It is correct when it lands exactly on the dashed target. For a rotation about another point, choose the rotation from the choices under the grid instead. Press New Challenge for a different challenge.
What to notice: the labeled vertices have to land on the matching labeled vertices of the target. A turn that puts the outline in the right place but the labels in the wrong places is a different rotation.
Six questions, one at a time. A first wrong answer earns a hint and one retry; the full explanation appears when a question is answered correctly or after a second miss. Your score counts first-try answers only.
Lesson 35 · Rotations · Mr. J Geometry. All activities on this page run entirely in your browser.