Isometries
Senior 2 · Mathematics
Isometries
Lesson 4 of 5
18 min
A sliding door, a bus moving along a straight stretch of the Kigali–Muhanga road, and a printing block moved along a strip of cloth all move without turning or flipping. Every point moves the same distance in the same direction. That movement is a translation, and you already have the tool to describe it: a vector.
The translation by the vector v = ab moves every point by v: for every point A, AA′ = v. With coordinates:
(x, y) → (x + a, y + b)
Worked example 1
Translate triangle A(1, 2), B(4, 2), C(2, 5) by v = 3−4.
Add 3 to every x and subtract 4 from every y: A′(4, −2), B′(7, −2), C′(5, 1).
Figure 4. Translation by the vector (3, −4): the three arrows AA′, BB′ and CC′ are equal vectors. Diagram: Ganzaa.
Worked example 2
A translation maps P(−2, 3) to P′(4, 1). Find its vector and the image of the origin.
v = PP′ = 4 − ( − 2)1 − 3 = 6−2. The origin goes to (0 + 6, 0 − 2) = (6, −2).
Tip
Translation = add the vector. To find the vector, use end minus start: v = image − object. To go backwards from an image to its object, subtract the vector.
Translate (−3, 4) by 5−6.
Solution: ( − 3 + 5, 4 − 6) = (2, −2).
Which vector translates (7, −1) onto (2, 3)?
Solution: 2 − 73 − ( − 1) = −54.
A point is translated by 21 and then by −53. Which single translation has the same effect? Where does the origin go?
Solution: 21 + −53 = −34; the origin goes to (−3, 4).
Under the translation by 4−2, the image of A is A′(1, 1). Find A.
Solution: subtract the vector: A = (1 − 4, 1 + 2) = ( − 3, 3).
Remember
Sources and credits
The Rwandan examples, practice items and exercises are written by Ganzaa.
