Rigid motions can be strung together: reflect, then translate, then rotate.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The result is still congruent
Each move preserves lengths and angles, so any number of them still does.
Order matters
Reflecting then translating usually lands somewhere different from translating then reflecting.
Working out a sequence
Match one corner first with a translation, then fix the orientation with a reflection or rotation.
Moves combine into one journey
Apply a translation and then a reflection and the overall effect is a single transformation, even if it does not have a simple name. Composing moves lets you get from any figure to any congruent copy of it.
Order usually matters
Reflecting then translating generally gives a different result from translating then reflecting. Transformations do not commute in general, so a description must state the order as well as the moves.
Two reflections make something else
Reflecting across two parallel lines produces a translation. Reflecting across two intersecting lines produces a rotation. Every rigid motion can in fact be built from reflections alone, which is a surprising and elegant fact.
A sequence is a proof of congruence
To show two figures are congruent, exhibit a sequence of rigid motions carrying one to the other. That is a complete argument, not an approximation — which is why this definition of congruence is the useful one.
Step 2: Try It Yourself
Tap and try it out.
Every length and every angle is exactly as it was. The shape moved without changing, so the two figures are congruent.
Not used by a reflection
Step 3: Watch an Example
One step at a time.
Watch Leo Find a Sequence
Leo must carry one triangle exactly onto another.
- Step 1
He picks a corner on each triangle that should match.
Step 4: Your Turn
Practice makes it stick.
Two Moves
Problem 1 of 2
A shape is translated 3 right, then 4 right. How far right in total?
Two Reflections
Problem 2 of 2
A shape is reflected across the same line twice. How many units from its start?
String Them Together
1 of 4
Do rigid motions keep side lengths the same? 1 for yes, 0 for no.
2 of 4
Translated 5 right then 2 left. Net movement right?
3 of 4
Are two figures congruent if a sequence of rigid motions maps one to the other? 1 for yes, 0 for no.
4 of 4
Rotated 90° then 90° again. Total turn in degrees?
Step 5: Quick Check
Show what you know.
Question 1 of 1
Rotated 90° three times. Total turn in degrees?
What You Learned
- Rigid motions can be applied one after another.
- The result is still congruent to the original.
- Two figures are congruent exactly when some sequence carries one onto the other.