A shape can be slid, flipped, or turned. Those are translation, reflection, and rotation.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Try it first
A triangle sits on a pair of axes. Its corner A is at (2, 1). The whole triangle slides 5 squares to the right. What is the first coordinate of A after the slide?
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
All three share one property.
Rigid motion is a move that changes a figure’s position but not its lengths or angles.
What does change
Only where the shape is, and which way it faces. Its size and shape are untouched.
Why this matters
Rigid motions are how congruence gets defined, and congruence is the whole of the next lesson.
Step 2: Try It Yourself
The dashed shape is the original. It stays so you can compare.
Every length and every angle is exactly as it was. The shape moved without changing, so the two figures are congruent.
Step 3: In Real Life
A phone screen
Rotate your phone and the photo turns; nothing in it changes. Drag an icon and it slides. Screens are built on rigid motions.
Step 4: Watch an Example
One step at a time.
Watch Maya Slide a Triangle 6 Right
Triangle ABC has corners at A(1, 2), B(4, 2) and C(1, 6). Maya slides the whole triangle 6 squares right, then checks what survived the move.
- Step 1
Each corner makes the same move, so 6 is added to every first coordinate. A(1, 2) becomes (7, 2), B(4, 2) becomes (10, 2), and C(1, 6) becomes (7, 6).
Watch Dev Flip a Point Across the Vertical Axis
Point P is at (3, 4). Dev reflects it across the vertical axis. He writes the first two steps; you supply the rest.
- Step 1
A reflection needs a mirror line. Here the mirror is the vertical axis: the line through 0 that runs up and down.
Watch Noor Turn a Triangle
Triangle PQR has a square corner at P. Noor turns the whole triangle a quarter turn about P.
- Step 1
A turn needs a center and an angle. Noor’s center is the corner P, and the angle is a quarter turn: 90°.
Step 5: Your Turn
Practice makes it stick.
The Fence
Problem 1 of 17
A fence has three identical panels in a row. Each one is the first panel slid along. One panel takes 14 feet of timber, and timber costs $3 a foot. What does the timber for all three panels cost, in dollars?
dollars
The Door Panes
Problem 2 of 17
A glazier cuts a glass pane with sides of 9, 12 and 15 inches. For the other side of the door she cuts its mirror image. Edging goes all the way around each pane and costs $0.25 an inch. What does the edging for both panes cost, in dollars?
dollars
Slide, Flip, Turn
Follow one point through the move, then read off what is asked.
Problem 3 of 17
A point at (1, 3) is translated 5 squares right. What is its new first coordinate?
Problem 4 of 17
A point at (6, 2) is reflected across the vertical axis. What is its new first coordinate?
Problem 5 of 17
A point at (3, 8) is reflected across the horizontal axis. What is its new second coordinate?
Problem 6 of 17
A rigid motion carries shape F onto shape G. One angle of F measures 40°. What does the matching angle of G measure, in degrees?
degrees
Problem 7 of 17
A point at (5, 2) is translated 8 squares left. What is its new first coordinate?
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, 2) across the vertical axis and got (5, −2). One line is a slip. Which?
Priya’s working
- 1(5, 2), mirror: the vertical axis
- 2the mirror is the vertical axis, so up and down swap
- 3second coordinate: 2 → −2
- Answer(5, 2) → (5, −2)
Problem 9 of 17
Fix Priya’s line 2 and finish. What is the first coordinate of the image of (5, 2)?
Problem 10 of 17
Leo slid triangle ABC 4 squares right. He says side AB grew from 6 units to 10 units. One line is a slip. Which?
Leo’s working
- 1AB = 6 units before the slide
- 2the slide moves every corner 4 right
- 3AB = 6 + 4 = 10 units after the slide
- Answerso the triangle is bigger after the slide
Problem 11 of 17
Fix Leo’s line 3. Side AB measured 6 units before the slide. How long is it after the slide, in units?
units
Problem 12 of 17
Sam reflected a shape across a line. One of its angles measured 35°. She says the matching angle in the image measures 145°. One line is a slip. Which?
Sam’s working
- 1the angle before the flip: 35°
- 2a flip turns the shape over, so the angle gets bigger
- 3180 − 35 = 145
- Answerthe matching angle is 145°
Problem 13 of 17
Fix Sam’s line 2. What does the matching angle in the image measure, in degrees?
degrees
Pick the Move
Decide which move is being described before you work anything out.
Problem 14 of 17
The letter R is drawn on clear plastic. The plastic is picked up, turned over and laid down again, and the R now reads backwards. Which move was that?
Problem 15 of 17
Triangle T sits above the horizontal axis. Its image must sit below that axis, with every corner the same distance from the axis as before. Which single move does that?
Same Size?
Select every option the question asks for. Nothing here needs working out.
Problem 16 of 17
Select every move that leaves a shape exactly the same size and shape.
Problem 17 of 17
Select every quantity that a slide, a flip or a turn must leave unchanged.
Step 6: Quick Check
Show what you know.
Question 1 of 2
(2, 9) reflected across the vertical axis. New first coordinate?
Question 2 of 2
What does a rigid motion preserve?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
A shape is reflected across the vertical axis, and one of its corners does not move at all. What must the first coordinate of that corner be?
Stretch 2 of 2
A shape is flipped across the vertical axis. The image is then flipped across the vertical axis again. Where does the shape end up?
What You Learned
- Translation slides, reflection flips, rotation turns.
- All three keep every length and every angle.
- Only position and facing change.
For the grown-up
The misconception this lesson targets is that moving a shape changes it: a figure slid further along, or flipped over, is quietly read as bigger, smaller or a different shape. Its cousin is the wrong axis. A flip across the vertical axis, the one that runs up and down, changes left and right and leaves the height alone; a child who has learned "reflection flips the sign" often flips the wrong coordinate. The cure for both is to follow one corner through the move and say out loud what it did.
Three moves that preserve everything — A translation slides a figure, a reflection flips it across a line, and a rotation turns it about a point. All three leave every length and every angle exactly as it was. They are called rigid motions because the figure behaves like a solid object being moved.
What does change — Position changes; orientation may change. A reflected figure faces the other way, which matters for letters and for hands but not for lengths. Distinguishing what a move preserves from what it alters is the whole content of this chapter.
Describing a move precisely — A translation needs a direction and a distance. A reflection needs the mirror line. A rotation needs a center, an angle and a direction. Naming the move is not enough — a complete description lets someone else reproduce it exactly.
Try it with tracing paper — Trace a shape, then slide, flip or turn the tracing. Whatever you can do to the tracing without stretching it is a rigid motion. That physical test is a reliable check on whether a transformation preserves shape and size.
Trace it and move the tracing — Trace the triangle onto tracing paper. Slide the tracing across. Every side is still the same length. Flip it over. Still the same lengths and angles. Turn it. Still the same. Anything you can do to the tracing without stretching it is a rigid motion.