Which of the following does a rigid motion preserve?

Rigid Motions: What Stays the Same When You Move Things Around?

Geometry’s full of ways to mess with shapes – stretching, squashing, you name it. But some transformations are a bit more…respectful. We call them rigid motions. Think of it like picking up a puzzle piece and moving it somewhere else on the table without bending it or anything. So, what exactly stays the same when you do that? Let’s take a look.

What’s a “Rigid Motion,” Anyway?

Okay, first things first: what is a rigid motion? Simple. It’s moving something without changing its size or shape. Imagine sliding a book across a table, spinning it around, or flipping it over. That’s the idea. No stretching, no shrinking, no funny business. There are basically three main moves in the rigid motion playbook: sliding (translation), turning (rotation), and flipping (reflection). Oh, and there’s a bonus move: the glide reflection. It’s just a flip and a slide combined.

• Translation: Imagine pushing a chess piece across the board. That’s a translation.
• Rotation: Think of a spinning ballerina. That graceful turn? Rotation.
• Reflection: Hold a picture up to a mirror. The image you see? Reflection.

The Good Stuff: What Doesn’t Change

Here’s the heart of the matter: what properties are “safe” when you perform a rigid motion? Well, quite a few, actually. The key is that rigid motions don’t distort anything.

• Distances Stay Put: This is huge. The distance between any two points on your shape? Stays exactly the same. This is really the foundation of it all.
• Angles? Untouched: Angles inside the shape don’t change, either. So, if you have a perfect right angle, it stays a perfect right angle.
• Sides Keep Their Length: Because distances are preserved, the sides of any shape within your figure don’t change length. Makes sense, right?
• Perimeter? Still the Same: Since the sides are the same, the distance around the shape is also the same.
• Area Doesn’t Budge: The amount of space the shape takes up remains constant.
• Shape and Size? Obviously!: This is the whole point! The figure isn’t distorted in any way. The original shape and the new shape are exactly the same, just in a different spot.
• Parallel Lines Stay Parallel: If you have parallel lines, they’ll stay parallel after any rigid motion. They won’t suddenly converge or diverge.
• Points on a Line Stay on a Line: If you have points lined up on a line, they’ll stay lined up on that line.

Basically, rigid motions are all about keeping the fundamental characteristics of a shape intact – its size and its overall form.

What Does Change?

Okay, so lots of things stay the same. But what can change?

• Location, Location, Location: The position of the shape definitely changes. That’s the whole point of moving it!
• Orientation Can Flip: Reflections, in particular, will change the orientation. It’s like seeing your mirror image – things are reversed. Rotations also change the way the shape is facing.
• New Coordinates: The coordinates of the points that make up the shape will change because you’ve moved it to a new spot on the grid.

Why Bother?

Why are rigid motions so important? Well, they’re tied to the idea of congruence. If you can move one shape onto another using only rigid motions, then the shapes are congruent – they’re exactly the same. It’s a super-precise way of saying two things are identical, even if they’re in different places or orientations.

Plus, rigid motions are used everywhere. Computer graphics, robotics, engineering…anywhere you need to move things around without changing them, rigid motions are your friend.

The Bottom Line

Rigid motions are a fundamental part of geometry. They let us move shapes around without distorting them. By understanding what stays the same – distances, angles, area, shape – we get a better handle on what it means for two things to be truly identical. And that’s pretty cool, if you ask me.