Translation

• Rigid Motion: A translation is classified as a rigid motion or isometry, meaning that the distance between any two points remains constant after the transformation. This ensures that the pre-image and the image are congruent figures.

• Pre-image and Image: The original figure before the shift is called the pre-image, while the resulting figure after the shift is called the image. Every point $P(x, y)$ in the pre-image is mapped to a unique point $P'(x', y')$ in the image.

• Vector Representation: Translations are often described using a translation vector $\vec{v} = \langle a, b \rangle$, where $a$ represents the horizontal shift (along the x-axis) and $b$ represents the vertical shift (along the y-axis).

• Orientation Preservation: Unlike reflections, translations preserve the orientation (or 'handedness') of a figure. If the vertices of a triangle are labeled clockwise in the pre-image, they will remain clockwise in the image.

• Parallelism of Paths: Every point in the figure moves along a path that is parallel to the paths of all other points. The segments connecting corresponding points (e.g., $AA'$, $BB'$, $CC'$) are all parallel and equal in length.

• Distance Invariance: The distance between any two points $A$ and $B$ in the pre-image is exactly equal to the distance between their corresponding points $A'$ and $B'$ in the image. This is why the transformation is called a 'rigid' motion.

• Angle Preservation: The measure of any angle in the pre-image is equal to the measure of the corresponding angle in the image. This property ensures that the shape of the figure is not distorted during the slide.

Coordinate Rule Method

• Step 1: Identify the coordinates of the vertices of the pre-image, $(x, y)$.
• Step 2: Determine the horizontal shift $h$ and vertical shift $k$ from the translation description or vector $\langle h, k \rangle$.
• Step 3: Apply the algebraic rule $T(x, y) = (x + h, y + k)$ to each vertex to find the new coordinates.

Vector Addition Method

• Step 1: Represent each point as a position vector $\vec{p}$.
• Step 2: Add the translation vector $\vec{v}$ to the position vector: $\vec{p'} = \vec{p} + \vec{v}$.
• Step 3: The resulting vector $\vec{p'}$ provides the coordinates of the image point.

• Verify Parallelism: In exam problems, you can verify a translation by checking if the segments connecting pre-image vertices to image vertices are parallel. If they are not parallel, the transformation is likely a rotation or a combination of motions.

• Check the Vector Signs: Always double-check the signs of the translation vector. A positive $h$ moves the figure right, a negative $h$ moves it left; a positive $k$ moves it up, and a negative $k$ moves it down.

• Distance Check: Use the distance formula to ensure the side lengths of the image match the pre-image. If the lengths change, you have likely performed a dilation instead of a translation.

• Slope Consistency: The slope of the segment connecting $A$ to $A'$ must be identical to the slope of the segment connecting $B$ to $B'$. This is a quick way to catch calculation errors in coordinate geometry.

• Partial Translation: A common mistake is applying the translation to only one vertex or only one coordinate (e.g., shifting $x$ but forgetting to shift $y$). Every single point of the figure must undergo the exact same shift.

• Confusing Direction: Students often confuse 'left/right' with the $y$-axis or 'up/down' with the $x$-axis. Remember that $x$ is horizontal and $y$ is vertical.

• Pre-image/Image Swap: Ensure you are moving from the pre-image to the image. If you calculate the vector from the image back to the pre-image, the signs will be reversed, leading to an incorrect transformation rule.