Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 8, Unit 3, lesson 6, coordinates in four quadrants. In which quadrant is the point (−3, 5)?
Answer
The second quadrant: left of the y-axis (x is negative) and above the x-axis (y is positive). - From Year 8, Unit 3, integers. Work out −7 + 4.
Answer
−3. Start at −7 and move 4 to the right. - From Year 8, Unit 3, integers. Work out 2 − 6.
Answer
−4. Take away 6 from 2 and you go past zero.
Words used in this lesson
- transformation
- a rule that moves every point of a shape to a new place. For example: slide, flip, turn, enlarge.
- pre-image
- the shape before it is moved. For example: triangle ABC.
- image
- the shape after it is moved; its corners have a dash. For example: A′B′C′.
- translation
- a slide: every point moves the same distance in the same direction. For example: 3 right and 2 down.
- vector
- the numbers that say how far across and how far up. For example: (3, −2).
One stamp, moved across a wall
A printer in Fez has a triangular stamp for a border pattern on a tiled wall. Its corners sit at A (−2, 1), B (1, 1) and C (−2, 3) on a grid of the wall. She slides the whole stamp 5 units right and 2 units down. The original is the pre-image; where it ends up is the image, and its corners are written A′, B′ and C′ (said “A prime”).
Predict first: where does corner A land? Write the coordinates, then use the sliders to test your answer with 5 across and −2 up.
A (−2, 1) moves 3 right and 2 down to A′ (1, −1). B and C move exactly the same way.
Show me the steps
- Across 3, up −2 (3 right, 2 down): A (−2, 1) goes to (1, −1), B (1, 1) to (4, −1) and C (−2, 3) to (1, 1).
- Across 0, up 0: nothing moves, and each image is the point itself: A (−2, 1), B (1, 1) and C (−2, 3) stay put. The vector (0, 0) is the translation that does nothing.
- Across −5, up 2 (5 left, 2 up): A goes to (−7, 3), B to (−4, 3) and C to (−7, 5).
- Across 6, up −6: A goes to (4, −5), B to (7, −5) and C to (4, −3). All three corners have moved the same 6 right and 6 down.
A translation slides every point of a shape the same distance in the same direction. It is described by one vector ab: a is the move across (right is positive) and b is the move up (up is positive). Every point follows the rule (x, y) → (x + a, y + b).
Why: because every point has the same move added, the gaps between points do not change. The image has the same size and shape, and each side stays parallel to the side it came from.
A transformation that keeps size and shape is a congruence transformation, or isometry. A translation is one.
Working backwards
Often you are given the picture and have to find the vector. Compare any one corner with its image: image minus pre-image, across and then up. Then test the same vector on a second corner.
Puzzle 1. Your guess (0, 0) is still 4 across and −3 up short of the target.
Show me the steps
- Puzzle 1 with the guess (0, 0): the guessed image sits on top of the pre-image, and the target is 4 across and −3 up further on.
- Puzzle 1 with the guess (4, −3): still to move across 0 and up 0, so every corner lands on the target.
- Puzzle 2: the target is 5 left and 2 up. The guess (−5, 2) lands on it; (−5, 3) overshoots by 1 up.
- Puzzle 3: the target is straight down. Across stays 0, and up must be −4.
The first number is always the move across. The vector 3−2 means 3 right and 2 down. It does not mean 3 up. Swapping the numbers gives −23, which is 2 left and 3 up: a different slide that sends a point somewhere else.
Worked examples
Translate P (−3, 4) by 5−6.
- Across: −3 + 5 = 2.
- Up: 4 + (−6) = −2.
- The image is P′ (2, −2).
Translate triangle (1, 2), (4, 2), (1, 5) by 3 left and 4 down. The vector is −3−4, so subtract 3 from every x and 4 from every y: (−2, −2), (1, −2), (−2, 1).
Find the vector that takes Q (2, −1) to Q′ (−4, 3).
- Across: −4 − 2 = −6.
- Up: 3 − (−1) = 3 + 1 = 4.
- The vector is −64: 6 left and 4 up. Check on the way back: 2 + (−6) = −4 and −1 + 4 = 3.
Your go. The first step is given. Translate R (−5, −2) by 46.
- Across: −5 + 4 = −1.
- Find the move up, then write the image R′.
Check the answer
Up: −2 + 6 = 4, so R′ is (−1, 4).
Check before you go on
1. A translation moves every point 4 units left and 3 units up. Which rule does it follow?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. The point P (3, −5) is translated by the vector (−7, 2). Its image P′ is (x, y). What is x?
3. The same point P (3, −5) is translated by the vector (−7, 2). What is y?
Level 2 · Building
4. A corner M (−4, 6) is moved by a translation to M′ (3, 1). The vector is (a, b). What is a?
5. A rangoli motif has corners at (1, 2), (4, 2) and (3, 5). It is slid so that the corner (4, 2) lands on (−1, −3). What is the y-coordinate of the image of the corner (3, 5)?
6. Which statement is true for every translation?
Level 3 · Stretch
7. A shape is translated by (−4, 3), and the image is then translated by (6, −8). One single translation does the same job, with vector (a, b). What is a + b?
8. A triangle PQR has P (2, 1), Q (6, 1) and R (2, 4). After a translation, P′ is at (−3, 5). What is the y-coordinate of R′?
Something to think about
In the second figure the target image is always the same shape as the pre-image. Could a translation ever make the image bigger, or turn it? Use a slider and a corner to explain what is going on.
One way to think about it
No. Every corner moves by the same vector, so the distance between any two corners does not change, and each side stays parallel to where it started. Only the position changes, which is why a translation is a congruence transformation: the image is an exact copy.
Exit ticket
9. Which rule is a translation?