Rigid motions can be strung together: reflect, then translate, then rotate.
Step 1: Picture It
Start with something you already know.
Socks and shoes
- Every morning you put on your socks, then your shoes.
- Try the same two steps the other way: shoes first, then socks.
- Now the socks end up over your shoes.
- Nothing new was added, yet changing the order changed the result.
A sequence of moves works the same way: each move starts where the last ended.
A flip then a slide can land somewhere different from a slide then a flip.
Where the picture stops: two slides land in the same place in either order.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
Point B is at (5, 3). It is reflected across the x-axis. What is the second coordinate of its image?
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
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.
Step 3: 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.
Step 4: In Real Life
A robot arm
A robot arm turns 90 degrees, then slides 10 centimeters. The part ends where the sequence puts it. Factories program motion as sequences of rigid moves.
Step 5: Watch an Example
One step at a time.
Watch Maya Do Two Moves in Order
Corner A of a shape sits at (4, 2). Maya reflects the shape across the y-axis, then translates it 6 right. She takes one move at a time.
- Step 1
The first move is the reflection. Across the y-axis, a point (x, y) goes to (−x, y).
Watch Dev Try the Same Two Moves the Other Way
Dev starts from the same corner, A(4, 2). He translates 6 right first, then reflects across the y-axis. He writes the first step; you supply the rest.
- Step 1
The first move now is the translation. Adding 6 to the first coordinate sends A(4, 2) to (10, 2).
Watch Noor Carry One Triangle onto Another
Triangle ABC and triangle DEF are the same size and shape. Noor looks for a sequence of moves that carries ABC exactly onto DEF.
- Step 1
Noor picks the pair of corners that should meet: A and D. A translation can be chosen to take any corner exactly onto its partner.
Step 6: Your Turn
Practice makes it stick.
The Doorway Border
Problem 1 of 17
A tiler makes a border beside a door. She lays one tile, then slides it along three more times to make a row of 4. For the other side of the door she flips the whole row across the doorway. Each tile costs $9. What do the tiles for both rows cost, in dollars?
dollars
The Window Stickers
Problem 2 of 17
A shop cuts a triangular window sticker with sides of 6, 8 and 10 inches. For the second sticker it turns the first design a half turn, then slides it along the window. Gold edging runs round all three sides of each sticker and costs $0.50 an inch. What does the edging for both stickers cost, in dollars?
dollars
One Move, Then the Next
Do the first move, write down where it lands, then do the second move from there.
Problem 3 of 17
A shape is translated 4 right, then translated 7 right. One slide would do the same job. How far right is it, in units?
units
Problem 4 of 17
A point at (2, 5) is reflected across the y-axis, then translated 3 up. What is its second coordinate at the end?
Problem 5 of 17
A point at (6, 3) is translated 2 left, then reflected across the y-axis. What is its first coordinate at the end?
Problem 6 of 17
A point at (3, 4) is turned a quarter turn counterclockwise about the origin, then translated 2 up. What is its second coordinate at the end?
Problem 7 of 17
A shape is reflected across the x-axis, then reflected across the x-axis again. One corner started at (7, 2). What is that corner’s second coordinate at the end?
Whose Step?
One line in each piece of working is a slip. Find it, then put it right.
Problem 8 of 17
Priya reflected the point (5, 1) across the y-axis and then translated it 4 right. She got (9, 1). One line is a slip. Which?
Priya’s working
- 1(5, 1), reflect across the y-axis, then translate 4 right
- 2reflect: (5, 1) → (−5, 1)
- 3translate 4 right: 5 + 4 = 9
- Answer(5, 1) → (9, 1)
Problem 9 of 17
Fix Priya’s line 3. Starting from (5, 1), reflect across the y-axis and then translate 4 right. What is the first coordinate at the end?
Problem 10 of 17
Leo translated the point (2, 1) five up and then reflected it across the x-axis. He says the two moves could have been done in either order. One line is a slip. Which?
Leo’s working
- 1(2, 1), translate 5 up, then reflect across the x-axis
- 2translate 5 up: (2, 1) → (2, 6)
- 3reflect across the x-axis: (2, 6) → (2, −6)
- Answerthe other order would land on (2, −6) as well
Problem 11 of 17
Fix Leo’s line 4. Starting from (2, 1), reflect across the x-axis first, then translate 5 up. What is the second coordinate at the end?
Problem 12 of 17
Sam flipped a triangle, then turned it, then slid it. Side AB measured 8 units at the start. She says three moves in a row must have shortened it. One line is a slip. Which?
Sam’s working
- 1AB = 8 units at the start
- 2a flip keeps every length, a turn keeps every length, a slide keeps every length
- 3three moves in a row must shorten it a little
- AnswerAB is under 8 units at the end
Problem 13 of 17
Fix Sam’s line 3. Sam’s triangle had sides of 8, 9 and 13 units at the start. What is the distance all the way round it after the flip, the turn and the slide, in units?
units
Pick the Move
Decide which sequence is being asked for before you work anything out.
Problem 14 of 17
Shape S is drawn here. It must finish below the x-axis and to the left of the y-axis. Every corner must stay the same distance from each axis as it is now. Which sequence does that?
Problem 15 of 17
Triangle ABC must be carried exactly onto triangle DEF. They are the same size and shape, and no corner of ABC is where its partner is. Which move makes sense first?
Same Place, Same Shape?
Select every option the question asks for. Nothing here needs working out on paper.
Problem 16 of 17
Select every sequence that finishes with the shape exactly where it started.
Problem 17 of 17
A sequence of rigid motions carries shape S exactly onto shape T. Select every sentence that must then be true.
Step 7: Quick Check
Show what you know.
Question 1 of 1
Rotated 90° three times. Total turn in degrees?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
A corner sits at (6, 2). Noor wants a sequence of two moves that finishes with that corner back at (6, 2) but the shape facing the other way. Which sequence does that?
Stretch 2 of 2
A corner starts at (2, 3). It is translated some distance to the right, then reflected across the y-axis, and it finishes at (−7, 3). How far right was the translation, in units?
units
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.
For the grown-up
The misconception this lesson targets is that the second move starts from the original position. A child who has flipped (5, 1) to (−5, 1) will often slide the 5 rather than the −5, because the 5 is the number written in the question. Its cousin is that order does not matter: the same flip and the same slide, swapped over, land in two different places. The cure for both is to write the in-between point down on its own line, and to read the next move off that line rather than off the question.
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.
Do them one at a time — Reflect across the y-axis, then translate 3 up. (2, 1) reflects to (−2, 1). Then translate: (−2, 4). Doing them in the other order gives a different result. Order matters, so apply them one at a time and write each step down.