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Identify Congruent Figures Using Rigid Motions

Students determine whether two figures are congruent by describing a sequence of rigid motions (translations, reflections, rotations) that maps one onto the other. They explain why two figures are NOT congruent if no such sequence exists.

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Identify Congruent Figures Using Rigid Motions

Grade 8 · Geometry: Transformations · question 1 of 3

0 of 20

20 rungs. A star at the top.

BC−8−6−4−22468−8−6−4−224680

Given segment BC is translated 3 up, type B′'s y-coordinate.

Perform the move.

How it is taught

0:00 of 4:25

Congruent figures match in shape and size.

A rigid motion slides, flips, or turns without resizing.

FGH−8−6−4−22468−8−6−4−224680

Read it: Congruent figures and rigid motions

Triangle UVW flips across the y-axis to U′V′W′. V(4, 2) lands at V′(−4, 2). UV is 3 long and so is U′V′: the flip moved the triangle without changing it, so the two are congruent.

  1. Name the move: a slide, a flip or a turn.
  2. Apply it to each vertex.
  3. Compare: every side and every angle of the image matches the original.

Slides, flips and turns are rigid motions: they move a figure without stretching it.

So a figure and its image under a rigid motion are always congruent.

Two figures are congruent when one rigid motion, or a sequence of them, carries one onto the other.

A rigid motion keeps every length and angle, so the image is congruent to the figure.

Thinking a flip makes the figure smaller or a different shape.

Measure a side before and after: UV and U′V′ are both 3. A flip only changes which way the figure faces.

Triangle ABC is reflected across the x-axis to make the congruent triangle A′B′C′. A is at (2, 6). What is the y-coordinate of A′? — −6. Across the x-axis, y changes sign: A′ has y = −6. The triangle only moved, so it is congruent.

Mapping △ABC onto △DEF with a translation and a reflection

Here are two triangles. The big question — are they congruent? Same shape, same size, just moved around?

To check, we use rigid motions: slides, flips, and turns. If a sequence of these maps one figure onto the other, they are congruent.

Let's start with a slide — a translation. We'll move the red triangle to the right, over to where the blue one lives.

After sliding, our triangle is in the right spot, but it's facing the wrong way.

So we flip it across a vertical line — a reflection.

Are these two triangles congruent?Rigid motions: slide · flip · turnFigure 1Figure 2Translation: slide to the rightSame place — wrong orientation.mirror lineReflection: flip across the mirror linePerfect match!RESULTCongruenttranslation + reflection
Grade 8 · Two triangles, three rigid motions, one big question

Look — it lines up perfectly on top of the blue triangle.

A translation followed by a reflection mapped the red triangle exactly onto the blue one, so the two figures are congruent.

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics
Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal
1 star — one skill mastered, every rung of its ladder climbed
Climbing now
Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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