Geometric Translation: Inverse Translations & Translation Vectors

8th Grade

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| By Thames

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| Questions: 20 | Updated: Dec 17, 2025

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1) Which statement best describes an inverse translation?

A reflection across the x-axis

A translation in the opposite direction of the original vector

A rotation by 180°

A scaling by a factor of –1

Inverse translations undo a movement by applying the same distance in the opposite direction. For a vector ⟨a, b⟩, the inverse is ⟨–a, –b⟩.

Explanation

Inverse translations undo a movement by applying the same distance in the opposite direction. For a vector ⟨a, b⟩, the inverse is ⟨–a, –b⟩.

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About This Quiz

Geometric Translation: Inverse Translations & Translation Vectors - Quiz

How can you reverse a translation or interpret the motion behind a vector? In this quiz, you’ll explore how translation vectors shift shapes and how inverse translations undo that movement. You’ll practice reading diagrams, comparing vector directions, and analyzing how coordinate changes reflect geometric motion. Step by step, you’ll strengthen... see moreyour understanding of how translations operate both forward and backward, gaining confidence in visualizing and calculating precise shifts on the coordinate plane.
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2) What first name or nickname would you like us to use?

You may optionally provide this to label your report, leaderboard, or certificate.

2) The rule (x, y) → (x + 4, y – 7) moves a point 4 units right and 7 units down. What rule represents its inverse?

(x,y)→(x−7,y+4)

(x,y)→(x+7,y−4)

(x,y)→(x−4,y+7)

(x,y)→(x−7,y−4)

To reverse a translation, switch the signs of both components: +4 → –4, –7 → +7. The inverse rule becomes (x, y) → (x – 4, y + 7).

Explanation

To reverse a translation, switch the signs of both components: +4 → –4, –7 → +7. The inverse rule becomes (x, y) → (x – 4, y + 7).

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3) If a translation vector is ⟨–6, 3⟩, what is its inverse?

⟨6,−3⟩

⟨−6,−3⟩

⟨3,−6⟩

⟨12,12⟩

Reverse both directions of the vector: ⟨–6, 3⟩ → ⟨6, –3⟩. Positive x means right, and negative y means down.

Explanation

Reverse both directions of the vector: ⟨–6, 3⟩ → ⟨6, –3⟩. Positive x means right, and negative y means down.

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4) A point M(4, –5) is translated by ⟨–3, 7⟩, then by its inverse. Where is the final image?

(1,2)

(4,−5)

(7,−12)

(−2,12)

Translation ⟨–3, 7⟩ → M′(1, 2). Applying ⟨3, –7⟩ brings it back: (1 + 3, 2 – 7) = (4, –5).

Inverse translations restore the original coordinates.

Explanation

Translation ⟨–3, 7⟩ → M′(1, 2). Applying ⟨3, –7⟩ brings it back: (1 + 3, 2 – 7) = (4, –5).

Inverse translations restore the original coordinates.

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5) The translation rule (x, y) → (x – 5, y + 2) is reversed by _______.

A translation is undone by performing the opposite movement in each direction, so adding 5 to x reverses the original leftward shift of 5 and subtracting 2 from y reverses the original upward shift of 2, restoring every point to its starting location.

Explanation

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6) Which of the following translation pairs are inverses?

(x+3,y+4) and (x+3,y−4)

(x−2,y+5) and (x+2,y−5)

(x−3,y+3) and (x+3,y+3)

(x+4,y+4) and (x+4,y−4)

The pair (x, y) → (x – 2, y + 5) and (x, y) → (x + 2, y – 5) is inverse because the second rule exactly cancels the first by shifting right what was shifted left and shifting down what was shifted up, returning every point to its original coordinates.

Explanation

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7) What happens when a translation and its inverse are applied successively to any point?

Moves twice as far

Returns to original position

Rotates

Reflects

Performing a translation followed by the exact opposite translation cancels the movement because each coordinate change is undone by an equal and opposite change, causing the point to end exactly where it began.

Explanation

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8) A triangle with vertices A(0, 0), B(2, 1), C(1, 3) is translated by ⟨4, –2⟩. What is the inverse vector to return it?

⟨−4,2⟩

⟨4,2⟩

⟨−2,4⟩

⟨2,−4⟩

Vectors and their inverses must point in opposite directions with equal magnitude, so reversing a shift of 4 units right and 2 units down requires moving 4 units left and 2 units up to exactly retrace the original motion.

Explanation

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9) A translation vector ⟨a, b⟩ is applied to a figure. What must be true about its inverse vector ⟨–a, –b⟩?

It changes orientation

Moves back to original position

Doubles movement

Flips figure

Because the inverse applies the opposite horizontal and vertical displacements, it undoes the effect of ⟨a, b⟩ by retracing the path in reverse, leaving the figure exactly where it started without altering its size, shape, or orientation.

Explanation

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10) The composite of ⟨2, –3⟩ followed by ⟨–2, 3⟩ results in which overall translation?

⟨4,−6⟩

⟨−4,6⟩

⟨0,0⟩

⟨2,−3⟩

Adding the vectors gives (2 + –2, –3 + 3) = (0, 0), meaning their combined effect is zero displacement, so the overall transformation is the identity and the figure does not move at all.

Explanation

Adding the vectors gives (2 + –2, –3 + 3) = (0, 0), meaning their combined effect is zero displacement, so the overall transformation is the identity and the figure does not move at all.

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11) A translation maps point P(3, –2) → P′(8, 1). Find the inverse rule.

(x+5,y+3)

(x−5,y−3)

(x−3,y−5)

(x+3,y−5)

The original translation moves points by ⟨5, 3⟩ because 8 – 3 = 5 and 1 – (–2) = 3, so its inverse must subtract 5 from x and subtract 3 from y to reverse the motion completely.

Explanation

The original translation moves points by ⟨5, 3⟩ because 8 – 3 = 5 and 1 – (–2) = 3, so its inverse must subtract 5 from x and subtract 3 from y to reverse the motion completely.

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12) The inverse of (x, y) → (x + a, y + b) is _______.

Undoing a translation requires applying the exact opposite shifts, meaning any added horizontal or vertical movement must be subtracted to send each point back to its initial coordinates.

Explanation

Undoing a translation requires applying the exact opposite shifts, meaning any added horizontal or vertical movement must be subtracted to send each point back to its initial coordinates.

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13) A shape is translated 9 units left and 4 units up. What is the inverse translation?

(x−9,y+4)

(x+9,y−4)

(x+4,y+9)

(x−4,y−9)

Since left corresponds to –9 and up corresponds to +4, reversing the translation requires shifting in the opposite directions—right 9 and down 4—to return the figure to its starting position.

Explanation

Since left corresponds to –9 and up corresponds to +4, reversing the translation requires shifting in the opposite directions—right 9 and down 4—to return the figure to its starting position.

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14) Which statement about inverse translations is true?

They change size

Reverse orientation

Preserve congruence and reverse movement

Distort shape

Inverse translations undo the movement of the original translation without changing shape, size, or orientation because translations are rigid motions, so applying one after the other retraces the path but does not distort the figure.

Explanation

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15) Which vector pair represents movements that cancel each other out?

⟨3,4⟩ & ⟨3,4⟩

⟨−2,5⟩ & ⟨−2,−5⟩

⟨4,−6⟩ & ⟨−4,6⟩

⟨−5,−3⟩ & ⟨5,−3⟩

Adding these vectors yields (4 + –4, –6 + 6) = (0, 0), showing they are perfect opposites and therefore move a point out and then back along the same path with no net displacement.

Explanation

Adding these vectors yields (4 + –4, –6 + 6) = (0, 0), showing they are perfect opposites and therefore move a point out and then back along the same path with no net displacement.

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16) What is the result of applying a translation vector ⟨a, b⟩ followed by ⟨–a, –b⟩?

Reflection

Rotation

Identity transformation

Dilation

The net movement becomes ⟨a + –a, b + –b⟩ = ⟨0, 0⟩, meaning the figure’s final position is identical to its original position, producing no visible change whatsoever.

Explanation

The net movement becomes ⟨a + –a, b + –b⟩ = ⟨0, 0⟩, meaning the figure’s final position is identical to its original position, producing no visible change whatsoever.

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17) Which vector is the inverse of ⟨0, –12⟩?

⟨12,−12⟩

⟨12,0⟩

⟨−12,0⟩

⟨0,12⟩

Since the original vector moves points 12 units downward, the inverse must move points 12 units upward, reversing the vertical displacement while leaving the horizontal motion unchanged at zero.

Explanation

Since the original vector moves points 12 units downward, the inverse must move points 12 units upward, reversing the vertical displacement while leaving the horizontal motion unchanged at zero.

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18) A figure undergoes translation (x, y) → (x + 7, y – 4). Applying which rule will return it to its start?

(x−7,y+4)

(x+4,y−7)

(x+7,y+4)

(x−4,y−7)

Reversing a translation simply reverses each coordinate adjustment, so undoing a shift of +7 in x and –4 in y requires subtracting 7 from x and adding 4 to y to restore each point’s original location.

Explanation

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19) If vector ⟨a, b⟩ moves figure F to F′, which vector moves F′ back to F?

⟨a,b⟩

⟨a,−b⟩

⟨−a,−b⟩

⟨−b,−a⟩

To return the figure to its original position, every movement must be undone by applying equal and opposite displacements, which only the vector ⟨–a, –b⟩ accomplishes.

Explanation

To return the figure to its original position, every movement must be undone by applying equal and opposite displacements, which only the vector ⟨–a, –b⟩ accomplishes.

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20) True or False: The inverse of ⟨0, 0⟩ is ⟨0, 0⟩.

True

False

Because ⟨0, 0⟩ represents no movement at all, reversing it requires another zero movement, making its inverse itself and demonstrating that the identity vector is self-inverse.

Explanation

Because ⟨0, 0⟩ represents no movement at all, reversing it requires another zero movement, making its inverse itself and demonstrating that the identity vector is self-inverse.

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