Geometry and Measurement - Geometric Transformations (Translation, Reflection, Rotation)

Transformation Basics: A transformation is a process that maps an original geometric figure, called the 'pre-image', onto a new figure called the 'image'. In Grade 7, we focus on isometric transformations (translations, reflections, and rotations) where the image remains congruent to the pre-image, meaning size and shape do not change. Visually, the image is often labeled with prime notation, such as point $A$ becoming $A'$.

Translation (Sliding): A translation moves every point of a figure the same distance in a specified direction without changing its orientation. Visually, it is like sliding a tile across a floor. It is often described using a column vector $\begin{pmatrix} a \\ b \end{pmatrix}$, where $a$ indicates horizontal movement (right is positive, left is negative) and $b$ indicates vertical movement (up is positive, down is negative).

Reflection (Flipping): A reflection flips a figure over a line called the 'axis of reflection' or 'mirror line'. Visually, the image is a mirror reflection of the pre-image. Every point and its image are equidistant from the line of reflection, and the line segment connecting them is perpendicular to the mirror line. Reflection changes the orientation of the shape (e.g., a 'right-facing' shape becomes 'left-facing').

Rotation (Turning): A rotation turns a figure around a fixed point called the 'center of rotation'. To describe a rotation, you need three pieces of information: the center of rotation (often the origin $(0,0)$), the angle of rotation (e.g., $90^\circ$, $180^\circ$), and the direction (clockwise or counter-clockwise). Visually, every point on the shape moves along a circular path around the center.

Lines of Reflection: Common mirror lines include the $x$-axis (the horizontal line where $y = 0$), the $y$-axis (the vertical line where $x = 0$), and diagonal lines like $y = x$ or $y = -x$. Visually, if you fold your graph paper along these lines, the pre-image will land perfectly on top of the image.

Properties of Isometry: Under translation, reflection, and rotation, the following properties are preserved (invariant): side lengths, angle measures, perimeter, and area. The only thing that changes is the position or orientation of the figure. Visually, the 'twin' shapes look identical in size and form, just placed differently on the coordinate plane.