Geometric Translation

1) 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)

Apply ⟨-4,-2⟩ to (0,0): x = 0+(-4) = -4, y = 0+(-2) = -2. New vertex = (-4,-2). Option B applies the vector to (2,1) incorrectly. Option C has no valid derivation. Option D applies the vector incorrectly to another vertex.

Explanation

Apply ⟨-4,-2⟩ to (0,0): x = 0+(-4) = -4, y = 0+(-2) = -2. New vertex = (-4,-2). Option B applies the vector to (2,1) incorrectly. Option C has no valid derivation. Option D applies the vector incorrectly to another vertex.

2) If Q(5,-3) → Q'(10,-3), what is the translation vector?

⟨5,0⟩

⟨-5,0⟩

⟨0,-5⟩

⟨0,5⟩

Δx = 10-5 = +5 and Δy = -3-(-3) = 0. Vector = ⟨5,0⟩, representing 5 units right with no vertical movement. Option B gives leftward motion. Options C and D give vertical movement, but y is unchanged in this translation.

Explanation

Δx = 10-5 = +5 and Δy = -3-(-3) = 0. Vector = ⟨5,0⟩, representing 5 units right with no vertical movement. Option B gives leftward motion. Options C and D give vertical movement, but y is unchanged in this translation.

3) Which vector moves a point 6 units left?

⟨6,0⟩

⟨-6,0⟩

⟨0,6⟩

⟨0,-6⟩

Leftward movement corresponds to a negative horizontal displacement. The vector ⟨-6,0⟩ describes a shift of 6 units in the negative x-direction with no vertical change. Option A moves 6 units right. Option C moves 6 units up. Option D moves 6 units down.

Explanation

Leftward movement corresponds to a negative horizontal displacement. The vector ⟨-6,0⟩ describes a shift of 6 units in the negative x-direction with no vertical change. Option A moves 6 units right. Option C moves 6 units up. Option D moves 6 units down.

4) 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, preserving orientation, distances between points, and area since no stretching or rotating occurs. However the position of the figure does change — it is in a different location after translation — making position the one property not preserved.

Explanation

Translations shift every point the same distance in the same direction, preserving orientation, distances between points, and area since no stretching or rotating occurs. However the position of the figure does change — it is in a different location after translation — making position the one property not preserved.

5) Which transformation is equivalent to two consecutive translations?

A single translation

Reflection

Rotation

Dilation

Two consecutive translations add their vectors: ⟨a1,b1⟩ followed by ⟨a2,b2⟩ = ⟨a1+a2, b1+b2⟩, which is itself a single translation. The composition of any number of translations is always another translation, never a reflection, rotation, or dilation.

Explanation

Two consecutive translations add their vectors: ⟨a1,b1⟩ followed by ⟨a2,b2⟩ = ⟨a1+a2, b1+b2⟩, which is itself a single translation. The composition of any number of translations is always another translation, never a reflection, rotation, or dilation.

6) 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)

Right 4 units adds 4 to x and down 6 units subtracts 6 from y, giving (x+4, y-6). Option A subtracts from x moving left. Option C adds to y moving up instead of down. Option D subtracts from x and adds to y, giving left and up.

Explanation

Right 4 units adds 4 to x and down 6 units subtracts 6 from y, giving (x+4, y-6). Option A subtracts from x moving left. Option C adds to y moving up instead of down. Option D subtracts from x and adds to y, giving left and up.

7) If M(-2,5) → M'(3,1), what vector was applied?

⟨-5,4⟩

⟨5,-4⟩

⟨-5,-4⟩

⟨5,4⟩

Δx = 3-(-2) = +5 and Δy = 1-5 = -4. Vector = ⟨5,-4⟩. Option A has wrong signs for both components. Option C has correct magnitude but wrong x sign. Option D has wrong y sign.

Explanation

Δx = 3-(-2) = +5 and Δy = 1-5 = -4. Vector = ⟨5,-4⟩. Option A has wrong signs for both components. Option C has correct magnitude but wrong x sign. Option D has wrong y sign.

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

Only squares

Only circles

All polygons

Only right triangles

Every shape remains congruent after translation because translations preserve all distances and angles. The rule applies to all figures — regular, irregular, simple, or complex. Options A, B, and D incorrectly restrict this property to specific shape types.

Explanation

Every shape remains congruent after translation because translations preserve all distances and angles. The rule applies to all figures — regular, irregular, simple, or complex. Options A, B, and D incorrectly restrict this property to specific shape types.

9) The image of 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)

Δx = -6-(-1) = -5 and Δy = 9-4 = +5. Rule = (x-5, y+5). Option B adds to x instead of subtracting. Option C gives wrong values. Option D gives wrong signs for both.

Explanation

Δx = -6-(-1) = -5 and Δy = 9-4 = +5. Rule = (x-5, y+5). Option B adds to x instead of subtracting. Option C gives wrong values. Option D gives wrong signs for both.

10) What is the translation rule for shifting square ABCD with A(0,0) to A'(3,1)?

(x+3, y-1)

(x-3, y+1)

(x+3, y+1)

(x-3, y-1)

Moving (0,0) to (3,1) requires adding 3 to x and adding 1 to y. The rule is (x,y) → (x+3, y+1). Option A subtracts 1 from y incorrectly. Option B subtracts from x. Option D subtracts from both coordinates.

Explanation

Moving (0,0) to (3,1) requires adding 3 to x and adding 1 to y. The rule is (x,y) → (x+3, y+1). Option A subtracts 1 from y incorrectly. Option B subtracts from x. Option D subtracts from both coordinates.

11) 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 without turning or flipping. The figure's shape, size, and orientation remain unchanged — only its position changes. Option A describes a reflection. Option C describes a rotation. Option D describes a dilation.

Explanation

A translation moves every point of a figure the same distance in the same direction without turning or flipping. The figure's shape, size, and orientation remain unchanged — only its position changes. Option A describes a reflection. Option C describes a rotation. Option D describes a dilation.

12) A translation moves point R(5,7) to R'(-2,10). What is the translation vector?

⟨7,3⟩

⟨-7,-3⟩

⟨-7,3⟩

⟨7,-3⟩

Δx = -2-5 = -7 and Δy = 10-7 = +3. Vector = ⟨-7,3⟩. Option A gives wrong sign for x. Option B gives wrong sign for y. Option D reverses both signs. Only option C correctly computes both components.

Explanation

Δx = -2-5 = -7 and Δy = 10-7 = +3. Vector = ⟨-7,3⟩. Option A gives wrong sign for x. Option B gives wrong sign for y. Option D reverses both signs. Only option C correctly computes both components.

13) Which vector represents 5 units right and 2 units down?

⟨-5,2⟩

⟨5,-2⟩

⟨2,5⟩

⟨-2,-5⟩

Rightward motion gives a positive x-component and downward motion gives a negative y-component, producing vector ⟨5,-2⟩. Option A moves left and up. Option C swaps the components. Option D moves left and down.

Explanation

Rightward motion gives a positive x-component and downward motion gives a negative y-component, producing vector ⟨5,-2⟩. Option A moves left and up. Option C swaps the components. Option D moves left and down.

14) If a figure is moved 8 units left and 2 units down, the translation rule is what?

(x+8, y+2)

(x-8, y+2)

(x+8, y-2)

(x-8, y-2)

Leftward movement subtracts from x and downward movement subtracts from y. The rule is (x,y) → (x-8, y-2). Option A adds in both directions, giving right and up. Option B subtracts x correctly but adds y incorrectly. Option C adds x incorrectly but subtracts y correctly.

Explanation

Leftward movement subtracts from x and downward movement subtracts from y. The rule is (x,y) → (x-8, y-2). Option A adds in both directions, giving right and up. Option B subtracts x correctly but adds y incorrectly. Option C adds x incorrectly but subtracts y correctly.

15) The translation rule (x,y) → (x-3, y+7) moves a point how?

3 right, 7 up

3 right, 7 down

3 left, 7 up

3 left, 7 down

Subtracting 3 from x shifts the point left along the horizontal axis, and adding 7 to y shifts it upward. The movement is 3 units left and 7 units up. Option A uses the wrong direction for x. Option B has both directions wrong. Option D has the y-direction wrong.

Explanation

Subtracting 3 from x shifts the point left along the horizontal axis, and adding 7 to y shifts it upward. The movement is 3 units left and 7 units up. Option A uses the wrong direction for x. Option B has both directions wrong. Option D has the y-direction wrong.

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

Translation ⟨a,b⟩

Translation ⟨-a,-b⟩

Reflection over x-axis

Rotation 180°

To reverse a translation, move the same distance in the opposite direction. The inverse of ⟨a,b⟩ is ⟨-a,-b⟩, which exactly cancels the original horizontal and vertical shifts. Option A repeats the original translation doubling the displacement. Options C and D are different types of transformations.

Explanation

To reverse a translation, move the same distance in the opposite direction. The inverse of ⟨a,b⟩ is ⟨-a,-b⟩, which exactly cancels the original horizontal and vertical shifts. Option A repeats the original translation doubling the displacement. Options C and D are different types of transformations.

17) 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: x' = 2+6 = 8 and y' = -5+4 = -1. New point A' = (8,-1). Option A subtracts 4 instead of adding. Options C and D subtract 6 from x instead of adding. Only option B correctly applies both additions.

Explanation

Apply the rule: x' = 2+6 = 8 and y' = -5+4 = -1. New point A' = (8,-1). Option A subtracts 4 instead of adding. Options C and D subtract 6 from x instead of adding. Only option B correctly applies both additions.

18) A translation can change a figure's orientation.

True

False

The answer is False. A translation slides a figure without flipping it. Orientation describes the facing direction of a figure, and since translations do not flip or rotate, the figure maintains the same orientation before and after. Reflections, not translations, are the transformation that reverses orientation.

Explanation

The answer is False. A translation slides a figure without flipping it. Orientation describes the facing direction of a figure, and since translations do not flip or rotate, the figure maintains the same orientation before and after. Reflections, not translations, are the transformation that reverses orientation.

19) Which property is preserved after a translation?

Distance between points

Angle measures

Shape and size

All of the above

Translations are rigid motions that preserve distances, angle measures, and orientation. Only the figure's position changes. Since all three properties listed in options A, B, and C are individually preserved, the correct answer is all of the above.

Explanation

Translations are rigid motions that preserve distances, angle measures, and orientation. Only the figure's position changes. Since all three properties listed in options A, B, and C are individually preserved, the correct answer is all of the above.

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

True

False

The answer is True. Translations are rigid motions that shift all points uniformly without stretching, compressing, or rotating. Side lengths and angles remain identical, so the image is always congruent to the pre-image. Any transformation that preserves distance and angle measure preserves congruence.

Explanation

The answer is True. Translations are rigid motions that shift all points uniformly without stretching, compressing, or rotating. Side lengths and angles remain identical, so the image is always congruent to the pre-image. Any transformation that preserves distance and angle measure preserves congruence.