Coordinate Rules for Rigid Motions

In this lesson:

• Find the coordinate rule for a translation
• Apply reflection rules across each axis
• Discover the three rotation rules about the origin

Learning Objectives for This Lesson

By the end of this lesson, you will be able to:

1. Apply the coordinate rule for any translation
2. Reflect a figure across each axis
3. Rotate a figure , , about the origin
4. Find an image by applying a rule to every vertex

Can You Figure Out the Rule?

Compare matching vertices. What happened to every x? To every y?

What Got Added to Each Coordinate?

Compare the original with its image .

• Each grew by : , ,
• Each grew by : , ,

Every point shifted by the same amounts — that's a slide.

The Translation Rule and Its Signs

A translation by across and up:

• Positive moves right, negative moves left
• Positive moves up, negative moves down

Apply It and Check a Side Length

Translate by left and down, so , .

• , and similarly for ,
• and — the side length didn't change

Deduce the Rule for a Mirror Image

This image is flipped, not slid. Compare the coordinates.

• , ,
• Which coordinate changed? Which stayed?

Reflection Rules Across Each Axis

Reflecting flips a figure across an axis like a mirror:

The coordinate that changes is the one not named in the axis.

Which Coordinate Actually Flips Its Sign?

It's tempting to flip the coordinate that matches the axis name — don't.

• Reflect across the -axis
• The point dropped from above to below — the changed
• The -axis is horizontal, so reflecting moves points vertically

Quick Check: Reflect a Point

Reflect the point across each axis. Commit before you advance.

• Across the -axis:
• Across the -axis:

Write both answers down first — then check yourself.

Turning Is Different From Sliding

Slides and flips keep coordinates in their lanes.

Rotation is the first move where the coordinates trade places.

Watch which coordinate becomes the new x, and which becomes the new y.

Discover the Quarter-Turn Coordinate Rule

Rotate a quarter-turn counterclockwise about the origin.

• , ,
• The new x is the old y, negated. The new y is the old x.

The 90-Degree Rule and a Distance Check

Rotation counterclockwise about the origin:

• Check
• Distance from origin: both times — preserved

The Half-Turn Rule About the Origin

Rotate the same triangle about the origin:

• , ,
• Both signs flip — the figure lands directly opposite the origin

The Three-Quarter Turn and the Table

Rotation Rules Hold About the Origin Only

These three rules work only when the center of rotation is the origin.

• A different center needs a different approach
• Always state the center: " counterclockwise about the origin"

If a problem rotates about another point, don't reach for these rules.

Identify the Rule From Coordinates

A figure maps so that for every vertex.

• and
• Both coordinates flipped sign on every point
• Which transformation is this? Name it precisely.

On Your Own: Rotate the Triangle

Rotate with vertices , , a quarter-turn counterclockwise about the origin.

• Apply the rule to each vertex yourself
• Plot the image and confirm the size didn't change
• No image row is given — build the whole thing.

Shift, Flip, and Swap the Coordinates

✓ Translation: — add to every coordinate

✓ Reflection: flip the sign of the coordinate not in the axis name

✓ Rotation: swap the coordinates (and negate) — about the origin only

Watch out: -axis reflection changes ; rotation rules need the origin as center

Coming Up: A Rule That Changes Size

Every rule today kept the figure the same size.

Next lesson introduces a move that multiplies the coordinates: dilation.

Multiplying is what finally makes a figure bigger or smaller.