pAIrmatic Lessons

Translations

8.G.1Grade 8Geometry20 practice questions

A translation slides every point of a figure the same distance in the same direction. Nothing turns, flips or stretches - only the position changes.

−1−111223344(2, 3)

Know these first

  • Master List skill 6.NS.6

This unlocks

What you'll be able to do

The big idea

A translation is a slide. Every point moves by the same amount across and the same amount up or down.

Write it as a rule: (x, y) → (x + a, y + b). Positive a moves right, negative a moves left; positive b moves up, negative b moves down.

If you know where one point lands, you know the whole rule: the shift is the image minus the original. Every other point gets the same shift.

Words you need

Image
Where a point or figure lands after a transformation, often labeled with a prime: A′.
Pre-image
The original figure before it moves.
Rigid motion
A move that keeps every length and angle: translations, reflections, rotations.
Orientation
The clockwise or counterclockwise order of a figure's corners.

Worked examples

Example 1

Translate the point (2, 3) using (x, y) → (x + 4, y − 1).

  1. Add 4 to x: 2 + 4 = 6.
  2. Subtract 1 from y: 3 − 1 = 2.
  3. The image is (6, 2).

Answer: (6, 2)

−1−111223344(2, 3)

Example 2

Triangle ABC has A(1, 1), B(3, 1) and C(1, 4). Slide it 2 units right and 3 units down.

  1. The rule is (x, y) → (x + 2, y − 3).
  2. A(1, 1) → (3, −2), B(3, 1) → (5, −2), C(1, 4) → (3, 1).

Answer: (3, −2), (5, −2), (3, 1)

−1−11122334455ABC

Example 3

Point (−1, 2) moves to (3, −1). Where does the point (0, 5) go under the same translation?

  1. The shift is image minus original: across 3 − (−1) = 4, up −1 − 2 = −3.
  2. The rule is (x, y) → (x + 4, y − 3).
  3. (0, 5) → (4, 2).

Answer: (4, 2)

−2−2−1−1112233445566(0, 5)(−1, 2)(3, −1)

Common mistakes

Moving x up and y sideways. x is the across value and y is the up-down value. "3 right" changes x; "3 up" changes y.

Subtracting in the wrong order when finding the rule. The shift is image minus original, not the other way around.

Thinking a translation can turn a figure. A slide never turns or flips anything. If the figure turned, it wasn't a translation.

Practice: 20 questions

12 of 20 are word problems. For a word problem, sketch it first and mark what you know.

Tier 1 Foundations Use the rule directly

  1. Translate the point (2, 5) using (x, y) → (x + 3, y − 4).
    −1−1112233445566(2, 5)
  2. Translate triangle ABC with A(0, 0), B(2, 0) and C(0, 3) using (x, y) → (x − 1, y + 2).
    −1−111223344ABC
  3. A game piece at (1, 4) moves 5 squares right and 2 squares down. Where does it land? Word problem
  4. A drone at (−3, 2) flies 4 units left and 6 units up. Where is it now? Word problem
  5. A bus at (2, −1) drives 1 block east (right) and 3 blocks north (up). Where is it now? Word problem

Tier 2 Intermediate One more step to think through

  1. Point (4, 1) moves to (1, 5), and point (6, 2) moves to (3, 6). What is the translation rule?
    −1−111223344556677ABA′B′
  2. Translate rectangle ABCD with A(1, 1), B(4, 1), C(4, 3) and D(1, 3) using (x, y) → (x − 5, y − 2).
    −1−11122334455ABCD
  3. A′(7, −2) is the image of A under (x, y) → (x + 3, y − 5). Find A.
    −3−3−2−2−1−11122334455667788(7, −2)
  4. A sticker on a map moves from (−2, −3) to (3, 1). What translation rule moved it? Word problem
  5. A sofa has corners at (2, 6) and (5, 6). It slides 4 units left and 1 unit down. Where do the corners end up? Word problem
  6. A robot starts at (0, 0) and moves by (x, y) → (x + 2, y + 3) three times in a row. Where does it end? Word problem
  7. A segment runs from (1, 2) to (4, 6). It is slid 7 units right. How long is the image segment? Word problem

Tier 3 Advanced Several steps - find the hidden structure

  1. Triangle ABC has A(−2, 1), B(0, 4) and C(1, −1). Its image has A′(1, −3), B′(3, 0) and C′(4, −5). What is the translation rule?
    −6−6−5−5−4−4−3−3−2−2−1−11122334455ABCA′B′C′
  2. The point (2, 5) is translated by (x, y) → (x + 2, y − 1), then by (x, y) → (x − 5, y + 4). Where does it end up? Word problem
  3. A puzzle piece slides so that its corner at (1, 1) lands on (6, −2). Where does the corner at (3, 4) land? Word problem
  4. A triangle with corners (0, 0), (3, 0) and (0, 4) is slid 6 units left and 2 units up. What is the perimeter of the image? Word problem
  5. Segment AB is translated so that A(−1, −1) lands on A′(2, 3). B is at (1, 2). Find B′.
    −2−2−1−111223344(1, 2)(−1, −1)(2, 3)

Tier 4 Challenge Stretch problems, some above grade level

  1. Which of these does a translation keep the same: side lengths, angle measures, orientation, position?
  2. A translation moves the point (2, −1) to (−3, 4). The image is then translated again by the same rule. Where does it end up? Word problem
  3. A rectangle with corners (1, 1), (5, 1), (5, 3) and (1, 3) is slid so that (1, 1) lands on (−2, 4). Where does (5, 3) land? Word problem

Answers

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