Geometry

Rigid Motions: Translate, Rotate, Reflect

A rigid motion (also called an isometry) moves every point of a figure the same way, preserving distances and angles. The three basic rigid motions are translation, rotation, and reflection. Let’s explore each one interactively!

Translation (Sliding)

A translation slides every point of a figure the same distance in the same direction. Think of it like dragging a shape on your screen without spinning or flipping it.

Below, the blue line is the original function y = x. Adjust the horizontal and vertical shift sliders to translate it.

Horizontal shift (h)0
-55
Vertical shift (k)0
-55
y=(x−0)+0y = (x - 0) + 0
-10-8-6-4-2246810-10-8-6-4-2246810Original: y = xTranslated
Try This

Try this: Set h = 3, k = 2. Every point on the original line moves right 3 and up 2. Notice the slope stays the same — translations don’t change shape or angle!

Translation of a Parabola

Let’s see the same idea with a parabola. Translations shift the vertex without changing the curve’s shape.

Horizontal shift (h)0
-55
Vertical shift (k)0
-55
y=(x−0)2+0y = (x - 0)^2 + 0
-8-6-4-22468-4-22468101214Original: y = x^2Translated parabola

Rotation

A rotation turns a figure around a fixed center point by a given angle. Below, we rotate the line y = x around the origin. The rotation angle changes the slope.

Rotation angle (degrees)0
0180
θ=0°New slope=tan⁡(45°+0°)\theta = 0°\quad \text{New slope} = \tan(45° + 0°)
-8-6-4-22468-8-6-4-22468Original: y = x (45 degrees)Rotated line
Try This

Try this: Set the rotation to 45 degrees. The original line (slope = 1, which is 45 degrees from horizontal) becomes vertical (slope = tan 90 degrees = undefined). Watch the line swing around!

Reflection

A reflection flips a figure over a line (called the line of reflection), creating a mirror image. The most common reflections are over the x-axis and y-axis.

Use the slider to blend between no reflection, reflection over the x-axis, and reflection over the y-axis.

Reflection (0=none, 1=x-axis, 2=y-axis)0
02
-8-6-4-22468-6-4-2246Original: y = 0.5x + 1Reflected

Reflecting a Parabola over the x-axis

A classic reflection: flipping y = x^2 over the x-axis gives y = -x^2.

a (1 = original, -1 = reflected)1
-11
y=1⋅x2y = 1 \cdot x^2
-6-4-2246-10-8-6-4-2246810Original: y = x^2y = a * x^2
Connection

Key idea: All three rigid motions preserve the size and shape of the figure. If two figures are related by any combination of translations, rotations, and reflections, they are congruent.
Challenge

Challenge: Can you describe a sequence of rigid motions that maps the point (1, 2) to (-1, -2)? (Hint: there are multiple correct answers!)
Take the Quiz