Geometric Translation

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.

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

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.

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.

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.

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.

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.

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

x change = 3 – (–2) = +5;

y change = 1 – 5 = –4.

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

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.

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.

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.

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.

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.

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.

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.

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⟩.

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.

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.

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.