A translation adds to the coordinates. Moving 3 right and 2 up sends (x, y) to (x + 3, y + 2).
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Try it first
Triangle ABC has corners A(2, 1), B(6, 1) and C(2, 4). The whole triangle is slid until A lands on (5, 3). Every corner makes the same move. What is the first coordinate of B 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.
Reflection changes one sign
Across the y-axis, (x, y) becomes (−x, y). Across the x-axis, it becomes (x, −y).
Rotation swaps them
A quarter turn counterclockwise about the origin sends (x, y) to (−y, x).
Check with one corner
Apply the rule to a single corner and see whether it lands where the picture says it should.
Step 2: Try It Yourself
The original stays on screen, so the movement is a relationship between two figures.
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 video game
A game moves a character 5 right and 2 up by adding to its coordinates: (x + 5, y + 2). Flipping it to face left negates x. Every game engine runs on these rules.
Step 4: Watch an Example
One step at a time.
Watch Sam Write the Rule for a Slide
A tile has a corner at P(2, 1) and another at Q(6, 1). Sam slides the whole tile 3 squares left and 4 squares up, and writes the rule that does it.
- Step 1
A slide to the left takes away from the first coordinate, and a slide up adds to the second. So this move sends (x, y) to (x − 3, y + 4).
Watch Dev Flip a Corner Two Ways
A corner D sits at (−6, 2). Dev flips it across the y-axis, then goes back and flips the original across the x-axis instead. He writes the first step; you supply the rest.
- Step 1
A flip across the y-axis changes the sign of the first coordinate. A flip across the x-axis changes the sign of the second. Dev starts with the y-axis.
Watch Noor Turn a Point Twice
Point T sits at (3, 1). Noor turns it a quarter turn counterclockwise about the origin, then turns the image again the same way.
- Step 1
A quarter turn counterclockwise about the origin sends (x, y) to (−y, x). The two coordinates swap places, and the new first one changes sign.
Step 5: Your Turn
Practice makes it stick.
The Moved Flower Bed
Problem 1 of 17
A plan shows a rectangular flower bed with corners at A(1, 2), B(7, 2), C(7, 5) and D(1, 5), measured in yards. The whole bed is slid 4 yards right and 2 yards up. Edging runs all the way round the bed and costs $5 a yard. What does the edging cost after the move, in dollars?
dollars
The Mirrored Bracket
Problem 2 of 17
A workshop cuts a steel bracket. On the plan its corners are P(2, 0), Q(10, 0) and R(2, 6), in inches. The longest side QR is 10 inches. For the other end of the shelf the same bracket is reflected across the y-axis. Steel edging runs along the two shorter sides of the reflected bracket at $2 an inch, and a protective cap on the longest side costs $9. What is the total, in dollars?
dollars
Apply the Rule
Name the rule first, then put the numbers into it.
Problem 3 of 17
(1, 6) is translated 4 squares right and 5 squares down. What is its new second coordinate?
Problem 4 of 17
(−7, 3) is reflected across the y-axis. What is its new first coordinate?
Problem 5 of 17
(2, −5) is reflected across the x-axis. What is its new second coordinate?
Problem 6 of 17
(4, 1) is turned a quarter turn counterclockwise about the origin. What is its new first coordinate?
Problem 7 of 17
(5, −3) is turned a quarter turn counterclockwise about the origin. What is its new second coordinate?
Whose Step?
One line in each piece of working is a slip. Find it, then put it right.
Problem 8 of 17
Leo slid a corner at (−2, 5) six squares left and two squares up. He wrote its image as (4, 7). One line is a slip. Which?
Leo’s working
- 1(−2, 5), moved 6 left and 2 up
- 2moving left adds to the first coordinate
- 3first coordinate: −2 + 6 = 4
- Answer(−2, 5) → (4, 7)
Problem 9 of 17
Fix Leo’s line 2. The corner (−2, 5) moves 6 squares left and 2 squares up. What is the first coordinate of its image?
Problem 10 of 17
Ravi reflected the corner (−3, 4) across the y-axis. He wrote its image as (3, −4). One line is a slip. Which?
Ravi’s working
- 1(−3, 4), mirror: the y-axis
- 2the rule is (x, y) → (−x, y)
- 3first coordinate: −3 → 3, second coordinate: 4 → −4
- Answer(−3, 4) → (3, −4)
Problem 11 of 17
Fix Ravi’s line 3. What is the second coordinate of the image of (−3, 4) across the y-axis?
Problem 12 of 17
Maya turned the corner (2, 5) a quarter turn counterclockwise about the origin. She wrote its image as (5, −2). One line is a slip. Which?
Maya’s working
- 1(2, 5), a quarter turn counterclockwise about the origin
- 2the rule is (x, y) → (y, −x)
- 3y = 5 and −x = −2
- Answer(2, 5) → (5, −2)
Problem 13 of 17
Fix Maya’s line 2. What is the first coordinate of the image of (2, 5) after a quarter turn counterclockwise about the origin?
Pick the Move
Read the rule backwards. Decide which move was used before working anything out.
Problem 14 of 17
A corner at (3, 8) lands on (−8, 3). Which of these moves does that?
Problem 15 of 17
A shape has a corner at (2, 7) that lands on (2, −7), and a corner at (4, 3) that lands on (4, −3). Which single move does that?
Same Landing Spot?
Select every option the question asks for.
Problem 16 of 17
Select every move that sends the point (4, 0) to (−4, 0).
Problem 17 of 17
A shape is translated 5 squares right and 3 squares up. Select every statement that must be true.
Step 6: Quick Check
Show what you know.
Question 1 of 1
(6, −2) translated 3 left. What is the new x?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
A point at (2, 7) is turned a quarter turn counterclockwise about the origin. Its image is turned again the same way, and that image is turned a third time. What is the second coordinate after the third turn?
Stretch 2 of 2
A corner at (3, 6) has to finish on (−6, 3). Priya says one quarter turn counterclockwise about the origin does it. Dev says a flip across the y-axis followed by a slide 3 left and 3 down does it too. Two moves in a row are allowed: the second one starts from where the first one finished. Which is right?
What You Learned
- A translation adds the same amount to every point.
- A reflection flips the sign of one coordinate.
- A quarter turn about the origin swaps the coordinates and flips one sign.
For the grown-up
The misconception this lesson targets is that a flip changes the sign of both coordinates. A child who has learned "a reflection flips the sign" applies it to the pair rather than to one number, and turns every mirror into (−x, −y). The cure is to name the mirror line out loud before touching the numbers: a fold along the up-and-down line moves a point sideways, so only the first coordinate can change, and a fold along the flat line moves it up or down. The second slip worth watching is the minus landing on the wrong coordinate in the quarter-turn rule — (y, −x) written where (−y, x) belongs — which is quickest to catch by plotting one corner and turning the page a quarter turn.
Translation adds to the coordinates — Sliding a figure 3 right and 2 up sends (x, y) to (x + 3, y + 2). Every point moves by the same amounts, which is exactly what makes the figure keep its shape. The rule is one addition per coordinate.
Reflection flips a sign — Reflecting across the x-axis sends (x, y) to (x, −y); across the y-axis it sends (x, y) to (−x, y). One coordinate keeps its value and the other becomes its opposite, which is precisely what crossing that axis means.
Rotation about the origin — A 90° counterclockwise turn about the origin sends (x, y) to (−y, x). A 180° turn sends it to (−x, −y). These rules can be discovered by plotting a point, turning the paper, and reading the new coordinates.
Why algebraic rules are worth having — Drawing works for one figure; a rule works for all of them and can be applied without a diagram. This is the move from geometry as drawing to geometry as computation, and it is what makes computer graphics possible.
Watch what happens to the numbers — Point (3, 5). Translate 2 right and 1 down. Add 2 to x, subtract 1 from y: (5, 4). Reflect (3, 5) across the x-axis: (3, −5). The y flips sign. Across the y-axis: (−3, 5). The x flips. Every rigid motion is a rule about the coordinates.