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Grade 10 · Rotations

Packaging studio: Verify a distance-preserving rotation

Packaging studio investigation: verify a distance-preserving rotation. Change the quantities, follow the motion, and explain the result using the displayed mathematical relationship.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Packaging studio: Verify a distance-preserving rotation. Current rotation: 0°. Current coordinate: (1.75, 0). Preserved radius: 1.75Rotate around zero without changing radius0°r = 1.75(1.75, 0) · equal axis scale
The rotation center is the origin and the initial point lies on the positive x-axis. Angles range from −180° to 180° and both axes have equal scale. This preserves distance but does not represent a translation.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Current rotation
0°
Current coordinate
(1.75, 0)
Preserved radius
1.75

HD animation studio

From experiment to screen.

Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.

Understand what you are seeing

The idea behind the motion.

A rotation around the origin changes a point's direction while preserving its distance from the center. Positive angles turn counterclockwise and negative angles clockwise. At a quarter turn, coordinate swapping and sign changes are special cases of the general sine-cosine rule. This investigation begins with distance from origin = 1.75; final signed angle = -45. Treat the named setting as a constructed classroom model, then explain which relationships would still hold if its numbers changed.

A relationship to keep

(r,0) ↦ (r cos θ, r sin θ)

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Represent the given quantities

    Begin at (r,0) on the positive horizontal axis, a distance r from the origin.

  2. STEP 2

    Follow the changing model

    Playback follows the signed angle around the circle. The faint initial radius provides a reference direction.

  3. STEP 3

    Check and explain the relationship

    Check the final coordinates and verify x²+y²=r², including negative and half-turn rotations.

Your turn to explain

Make a prediction. Test your reasoning.

Packaging studio: Rotate (1.75,0) through -45° about the origin. Find the image.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

(1.237, -1.237). Its distance from the origin is still 1.75.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 10 · Geometry

Derive a rotation from two intersecting reflections

Explain why two reflections in intersecting lines produce a rotation by twice their oriented angle.