Geometric Translation

1) Which of the following best describes a translation?

A figure is flipped over a line

A figure slides without turning

A figure is rotated around a point

A figure is enlarged

A translation moves every point of a figure the same distance in the same direction. It is often called a “slide.” Unlike a rotation or reflection, the figure does not turn or flip—its shape and orientation stay the same.

Explanation

A translation moves every point of a figure the same distance in the same direction. It is often called a “slide.” Unlike a rotation or reflection, the figure does not turn or flip—its shape and orientation stay the same.

2) Which property is preserved after a translation?

Distance between points

Angle measures

Shape and size

All of the above

Translations are rigid motions—they preserve distances, angle measures, and orientation. Only the figure’s position changes.

Explanation

Translations are rigid motions—they preserve distances, angle measures, and orientation. Only the figure’s position changes.

3) If point A (2, –5) is translated by rule (x, y) → (x + 6, y + 4), what are the coordinates of A′?

(8,−9)

(8,−1)

(−4,−1)

(−4,−9)

Apply the rule to each coordinate:

x′ = 2 + 6 = 8 y′ = –5 + 4 = –1.

New point A′ = (8, –1). Translations simply add or subtract from each coordinate.

Explanation

Apply the rule to each coordinate:

x′ = 2 + 6 = 8 y′ = –5 + 4 = –1.

New point A′ = (8, –1). Translations simply add or subtract from each coordinate.

4) The translation rule (x, y) → (x – 3, y + 7) moves a point:

3 right, 7 up

3 right, 7 down

3 left, 7 up

3 left, 7 down

Because translating a point involves adjusting its coordinates directly, subtracting a value from x shifts the point left along the horizontal axis while adding a value to y shifts it upward along the vertical axis, so applying the rule results in a movement of exactly 3 units to the left and 7 units upward as the point relocates to its new translated position.

Explanation

Because translating a point involves adjusting its coordinates directly, subtracting a value from x shifts the point left along the horizontal axis while adding a value to y shifts it upward along the vertical axis, so applying the rule results in a movement of exactly 3 units to the left and 7 units upward as the point relocates to its new translated position.

5) Which vector represents a movement of 5 units right and 2 units down?

⟨−5,2⟩

⟨5,−2⟩

⟨2,5⟩

⟨−2,−5⟩

Rightward motion corresponds to a positive x-component and downward motion corresponds to a negative y-component, so the vector must be ⟨5, –2⟩ because it encodes exactly a horizontal increase of 5 units and a vertical decrease of 2 units, matching the described translation.

Explanation

Rightward motion corresponds to a positive x-component and downward motion corresponds to a negative y-component, so the vector must be ⟨5, –2⟩ because it encodes exactly a horizontal increase of 5 units and a vertical decrease of 2 units, matching the described translation.

6) A triangle with vertices (0, 0), (2, 1), and (1, 3) is translated by ⟨–4, –2⟩. Which is one vertex after translation?

(-4,-2)

(-2,-1)

(3,-2)

(-2,1)

A translation is applied by adding the vector to each coordinate, and applying ⟨–4, –2⟩ to the vertex (0, 0) gives (0 + –4, 0 + –2) = (–4, –2), showing that every vertex of the triangle shifts left 4 units and down 2 units to maintain the triangle’s shape and orientation.

Explanation

A translation is applied by adding the vector to each coordinate, and applying ⟨–4, –2⟩ to the vertex (0, 0) gives (0 + –4, 0 + –2) = (–4, –2), showing that every vertex of the triangle shifts left 4 units and down 2 units to maintain the triangle’s shape and orientation.

7) The image of point P (–1, 4) is P′ (–6, 9). What translation rule was applied?

(x−5,y+5)

(x+5,y+5)

(x−7,y−5)

(x+7,y−5)

Change in x = –6 – (–1) = –5; change in y = 9 – 4 = +5.

Rule = (x, y) → (x – 5, y + 5). Negative x means left; positive y means up.

Explanation

Change in x = –6 – (–1) = –5; change in y = 9 – 4 = +5.

Rule = (x, y) → (x – 5, y + 5). Negative x means left; positive y means up.

8) Which figure will remain congruent to its pre-image after translation?

Only squares

Only circles

All polygons

Only right triangles

Every shape—regular or irregular—remains congruent after a translation because translations preserve distances and angles.

Explanation

Every shape—regular or irregular—remains congruent after a translation because translations preserve distances and angles.

9) If M (–2, 5) → M′ (3, 1), which vector was applied?

⟨−5,4⟩

⟨5,−4⟩

⟨−5,−4⟩

⟨5,4⟩

x change = 3 – (–2) = +5;

y change = 1 – 5 = –4.

Thus the translation vector is ⟨5, –4⟩.

Explanation

x change = 3 – (–2) = +5;

y change = 1 – 5 = –4.

Thus the translation vector is ⟨5, –4⟩.

10) Which translation rule shifts a figure 4 units right and 6 units down?

(x−4,y−6)

(x+4,y−6)

(x+4,y+6)

(x−4,y+6)

A horizontal shift of 4 units to the right requires adding 4 to every x-coordinate, while a vertical shift of 6 units downward requires subtracting 6 from every y-coordinate, so the correct rule is (x, y) → (x + 4, y – 6) because it applies precisely those directional coordinate changes.

Explanation

A horizontal shift of 4 units to the right requires adding 4 to every x-coordinate, while a vertical shift of 6 units downward requires subtracting 6 from every y-coordinate, so the correct rule is (x, y) → (x + 4, y – 6) because it applies precisely those directional coordinate changes.

11) Which transformation undoes a translation by vector ⟨a, b⟩?

Translation ⟨a,b⟩

Translation ⟨−a,−b⟩

Reflection x-axis

Rotation 180°

To reverse any translation, one must move the exact same distance but in the opposite direction, meaning the inverse operation is another translation using vector ⟨–a, –b⟩, which perfectly cancels the horizontal and vertical shifts of the original movement.

Explanation

To reverse any translation, one must move the exact same distance but in the opposite direction, meaning the inverse operation is another translation using vector ⟨–a, –b⟩, which perfectly cancels the horizontal and vertical shifts of the original movement.

12) Which transformation is equivalent to two consecutive translations?

A single translation

Reflection

Rotation

Dilation

Because translations add their component vectors, performing ⟨a₁, b₁⟩ followed by ⟨a₂, b₂⟩ results in a single translation ⟨a₁ + a₂, b₁ + b₂⟩, showing that multiple translations combine into one overall shift rather than producing reflections, rotations, or dilations.

Explanation

Because translations add their component vectors, performing ⟨a₁, b₁⟩ followed by ⟨a₂, b₂⟩ results in a single translation ⟨a₁ + a₂, b₁ + b₂⟩, showing that multiple translations combine into one overall shift rather than producing reflections, rotations, or dilations.

13) Which of these would NOT be preserved by a translation?

Orientation

Distance

Area

Position

Translations shift every point the same distance in the same direction, so although orientation, distance, and area remain unchanged as part of rigid-motion behavior, the actual position of the figure does change, making position the one attribute not preserved.

Explanation

Translations shift every point the same distance in the same direction, so although orientation, distance, and area remain unchanged as part of rigid-motion behavior, the actual position of the figure does change, making position the one attribute not preserved.

14) Which vector moves a point 6 units left?

⟨6,0⟩

⟨−6,0⟩

⟨0,6⟩

⟨0,−6⟩

Movement to the left corresponds to a negative horizontal displacement, so the vector ⟨–6, 0⟩ correctly represents a shift of 6 units in the negative x-direction with no vertical movement.

Explanation

Movement to the left corresponds to a negative horizontal displacement, so the vector ⟨–6, 0⟩ correctly represents a shift of 6 units in the negative x-direction with no vertical movement.

15) If Q (5, –3) → Q′ (10, –3), what is the vector?

⟨5,0⟩

⟨−5,0⟩

⟨0,−5⟩

⟨0,5⟩

Computing the translation requires finding the change in each coordinate, with Δx = 10 − 5 = 5 and Δy = –3 − (–3) = 0, giving vector ⟨5, 0⟩, which describes moving 5 units to the right with no vertical change.

Explanation

Computing the translation requires finding the change in each coordinate, with Δx = 10 − 5 = 5 and Δy = –3 − (–3) = 0, giving vector ⟨5, 0⟩, which describes moving 5 units to the right with no vertical change.

16) If a figure is moved 8 units left and 2 units down, the rule is _______.

Leftward movement subtracts 8 from x and downward movement subtracts 2 from y, so the translation rule (x, y) → (x – 8, y – 2) correctly encodes the specified horizontal and vertical displacements applied to every point.

Explanation

Leftward movement subtracts 8 from x and downward movement subtracts 2 from y, so the translation rule (x, y) → (x – 8, y – 2) correctly encodes the specified horizontal and vertical displacements applied to every point.

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17) A translation moves point R (5, 7) to R′ (–2, 10). The vector is _______.

The translation vector is obtained by subtracting original coordinates from the image coordinates, giving Δx = –2 – 5 = –7 and Δy = 10 – 7 = 3, so the movement is accurately represented by ⟨–7, 3⟩.

Explanation

The translation vector is obtained by subtracting original coordinates from the image coordinates, giving Δx = –2 – 5 = –7 and Δy = 10 – 7 = 3, so the movement is accurately represented by ⟨–7, 3⟩.

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18) Translate square ABCD with A (0, 0) to A′ (3, 1). The translation rule is _______.

Shifting (0, 0) to (3, 1) requires adding 3 to x and adding 1 to y, so the entire figure must undergo the rule (x, y) → (x + 3, y + 1), which slides all points 3 units right and 1 unit up.

Explanation

Shifting (0, 0) to (3, 1) requires adding 3 to x and adding 1 to y, so the entire figure must undergo the rule (x, y) → (x + 3, y + 1), which slides all points 3 units right and 1 unit up.

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19) Translations preserve congruence because they keep side lengths and angles unchanged.

True

False

Translations are classified as rigid motions because they shift all points uniformly without stretching, compressing, or rotating the figure, meaning side lengths and angles remain identical, and thus congruence is fully preserved.

Explanation

Translations are classified as rigid motions because they shift all points uniformly without stretching, compressing, or rotating the figure, meaning side lengths and angles remain identical, and thus congruence is fully preserved.

20) A translation can change a figure’s orientation.

True

False

Translations slide a figure without flipping it, so whereas reflections reverse orientation, translations maintain the same facing direction and therefore leave orientation unchanged regardless of the movement.

Explanation

Translations slide a figure without flipping it, so whereas reflections reverse orientation, translations maintain the same facing direction and therefore leave orientation unchanged regardless of the movement.