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Rigid Motions Preserve Lines and Lengths

Rigid Motions Preserve Lines and Lengths

You will be able to:

  • Show that a straight line stays straight under any rigid motion
  • Verify that a segment keeps its exact length when slid, flipped, or turned
  • Explain why dilation is not a rigid motion
Grade 8 Math | 8.G.A.1.a
Rigid Motions Preserve Lines and Lengths

Learning Objectives for This Lesson

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

  1. Demonstrate that a line's image under any rigid motion is still a straight line
  2. Verify by calculation that a segment and its image have the same length
  3. Explain why rigid motions are called distance-preserving
  4. Distinguish rigid motions from dilation by checking length
  5. Predict and verify images of segments under specific rigid motions
Grade 8 Math | 8.G.A.1.a
Rigid Motions Preserve Lines and Lengths

Can Moving a Line Bend or Shorten It?

Picture a straight line drawn on tracing paper. Now:

  • Slide it · Flip it · Turn it

Two questions worth predicting before we test:

  • Could any of these bend the line into a curve?
  • If you measured a segment first, would the number change?
Grade 8 Math | 8.G.A.1.a
Rigid Motions Preserve Lines and Lengths

Tracing Paper: Slide, Flip, Turn

Slide, flip, and turn a ruled line — straight in every case

  • After each move, the traced line is still perfectly straight
Grade 8 Math | 8.G.A.1.a
Rigid Motions Preserve Lines and Lengths

Straight Stays Straight on the Grid

Five points on a line translate together onto a new straight line

  • Five points on , each shifted by
  • The images are still collinear — a straight line
Grade 8 Math | 8.G.A.1.a
Rigid Motions Preserve Lines and Lengths

A Horizontal Line Rotated Becomes Vertical

Line through and — a horizontal line.

Rotate counterclockwise about the origin:

  • and — the image is the vertical line

The direction changed. The straightness did not.

Grade 8 Math | 8.G.A.1.a
Rigid Motions Preserve Lines and Lengths

Quick Check: Lines and Curves

True or false?

A reflection can turn a straight line into a curved line.

Decide before the answer appears.

  • False — reflection, like every rigid motion, keeps lines straight
Grade 8 Math | 8.G.A.1.a
Rigid Motions Preserve Lines and Lengths

A Harder Question: Does the Length Change?

We have settled the shape: lines stay straight.

Now a different question about the number:

  • Measure a segment. Slide, flip, or turn it. Is it still the same length?
  • Straightness was easy to see — length is not obvious
Grade 8 Math | 8.G.A.1.a
Rigid Motions Preserve Lines and Lengths

Why the Length Cannot Change

A translation adds the same amount to both endpoints.

Both endpoints shift equally; the gaps that feed length are unchanged

  • The differences and stay the same — so the length does
Grade 8 Math | 8.G.A.1.a
Rigid Motions Preserve Lines and Lengths

Translation Keeps the Segment Length Fixed

Segment to :

Translate by to , :

Grade 8 Math | 8.G.A.1.a
Rigid Motions Preserve Lines and Lengths

Reflection Flips It — Length Still 5

Reflect the same across the -axis: , .

  • Orientation flipped; the squared terms erase the sign change
Grade 8 Math | 8.G.A.1.a
Rigid Motions Preserve Lines and Lengths

Every Rigid Motion: Same Length

Table: translation, reflection, 180 rotation all give length 5

  • Same segment, three rigid motions, one constant: length
Grade 8 Math | 8.G.A.1.a
Rigid Motions Preserve Lines and Lengths

Worked Example: Diagonal Segment CD

to . Find , then reflect across the -axis.

Reflection: , :

Grade 8 Math | 8.G.A.1.a
Rigid Motions Preserve Lines and Lengths

Your Turn: Rotate and Measure

Segment to . First find .

Now apply a rotation about the origin: , .

  • Compute — does it match ?
Grade 8 Math | 8.G.A.1.a
Rigid Motions Preserve Lines and Lengths

Predict the Length Before You Compute

A segment has length . It is reflected across a line, then rotated.

On your own, write down the length of the final image.

  • Answer: — each rigid motion preserves the length
Grade 8 Math | 8.G.A.1.a
Rigid Motions Preserve Lines and Lengths

Is That True of ALL Transformations?

Every rigid motion we tested kept the length fixed.

But "rigid" is a specific category. Let's test a transformation that is not rigid:

  • A dilation — does it keep segments the same length?
Grade 8 Math | 8.G.A.1.a
Rigid Motions Preserve Lines and Lengths

Dilation: Still Straight, but Longer

Segment to , length .

Dilate from the origin by scale factor : , .

The image is still straight — but the length tripled.

Grade 8 Math | 8.G.A.1.a
Rigid Motions Preserve Lines and Lengths

Sorting Transformations With Two Tests

Classification table: rigid motions versus dilation on two properties

  • Rigid motions pass both tests; dilation fails the length test
Grade 8 Math | 8.G.A.1.a
Rigid Motions Preserve Lines and Lengths

On Your Own: Verify a Rigid Motion

Segment to .

Apply the reflection across the -axis. With no image pre-drawn:

  • Find and
  • Compute and , and state whether length was preserved
Grade 8 Math | 8.G.A.1.a
Rigid Motions Preserve Lines and Lengths

Three Common Traps to Avoid

  • Moved means changed. Position and orientation change; length does not — that is what rigid means.
  • Reflection looks shorter. Flipping reverses orientation, never measure — squaring erases the sign.
  • Dilation is rigid. Straight is not enough; rigid requires same length too.
Grade 8 Math | 8.G.A.1.a
Rigid Motions Preserve Lines and Lengths

What Stays Fixed Under a Rigid Motion

A rigid motion can move a figure anywhere and turn it any way, yet it locks in two things:

  • A straight line stays straight
  • Every segment keeps its exact length — in any direction

That lock is what "rigid" means.

Grade 8 Math | 8.G.A.1.a
Rigid Motions Preserve Lines and Lengths

What's Next: Do Angles Survive Too?

You have shown rigid motions preserve straightness and length.

But a figure also has corners.

  • Next lesson: does a rigid motion keep every angle the same measure?
Grade 8 Math | 8.G.A.1.a