Translations and rigid transformations

About this lesson

Grade 8 › Rigid transformations and congruence

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Translations and rigid transformations

Discover

A triangle slides 3 units right and 2 units down without turning. What happens to its side lengths?

  1. They stay the same
  2. They grow by 3

Key idea

A translation moves every point by the same directed distance.

A translation by (a,b)(a, b) sends (x,y)(x, y) to (x+a,y+b)(x + a, y + b). Translations, reflections and rotations are rigid transformations: they preserve lengths and angle measures.

Worked example

Translate A(2,−1)A(2, -1) by 4 left and 3 up.

  1. Step 1Start at A(2, −1).
  2. Step 2a = −4: move 4 to the left.
  3. Step 3b = 3: move 3 up.
  4. Step 4The image is A′(2 − 4, −1 + 3) = (−2, 2).

a=−4a = -4, b=3b = 3
Left is negative x.
A′=(2−4,−1+3)A' = (2 - 4, -1 + 3)
Add.
A′=(−2,2)A' = (-2, 2)
Compute.

Your turn, guided

Your turn: the y-coordinate of (5,1)(5, 1) translated 6 down.

−5-5
1 − 6.

Practice ★

  1. Translate the point A(−4,−4)A(-4, -4) by 2 units right and 1 units down. Tap its image A′A'.
  2. Translate the point A(1,0)A(1, 0) by 4 units right and 4 units down. Tap its image A′A'.
  3. Translate the point A(2,−1)A(2, -1) by 1 units right and 2 units down. Tap its image A′A'.

Watch out!

Practice ★★

  1. A translation takes (−3,0)(-3, 0) to (0,5)(0, 5). By how many units does it move points horizontally (positive = right)?
  2. A translation takes (−2,4)(-2, 4) to (−4,4)(-4, 4). By how many units does it move points horizontally (positive = right)?
  1. A polygon with 9 sides is translated. What changes?
    1. Only its position: lengths and angles stay the same
    2. Its side lengths
    3. Its angles
  1. Translate triangle ABC 2 right and 2 down. Tap the images A′, B′ and C′.

Exit ticket

Three questions to check you have got it.

  1. Translate the point A(−4,−4)A(-4, -4) by 2 units right and 1 units down. Tap its image A′A'.
  2. Translate the point A(1,0)A(1, 0) by 4 units right and 4 units down. Tap its image A′A'.
  3. Translate the point A(2,−1)A(2, -1) by 1 units right and 2 units down. Tap its image A′A'.

Summary card

Translations

(x,y)→(x+a,y+b)(x, y) \to (x + a, y + b); rigid: lengths and angles kept.

Watch out: left and down are negative.

End of the lesson

Next lesson: Reflections in the coordinate plane Back to the chapter

Answers: Translations and rigid transformations

A

  1. (−2,−5)(-2, -5)
  2. (5,−4)(5, -4)
  3. (3,−3)(3, -3)

B

  1. 33
  2. −2-2

C

  1. Only its position: lengths and angles stay the same

D

  1. (−3,−2), (0,−2), (−2,0)(-3, -2),\ (0, -2),\ (-2, 0)

E

  1. (−2,−5)(-2, -5)
  2. (5,−4)(5, -4)
  3. (3,−3)(3, -3)

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