Lesson 1: Rigid Transformations

Property

A rigid transformation is a change in the position of a figure that preserves its size and shape. The three main types of rigid transformations are:

1. Translations (slides)
2. Reflections (flips)
3. Rotations (turns)

Examples

Property

A translation is a rigid motion of the plane that moves horizontal lines to horizontal lines and vertical lines to vertical lines.

• A translation preserves the lengths of line segments and the measures of angles.
• For a translation, there is a pair $(a, b)$, called the vector of the translation, such that the image of any point $(x, y)$ is the point $(x + a, y + b)$.
• Under a translation, the image of a line $L$ is a line $L'$ parallel to $L$. Furthermore, translations take parallel lines to parallel lines. This is because a translation does not change the slope of a line.
• A translation that does not leave every point fixed does not leave any point fixed.

Examples

• The point $P(2, 5)$ is translated by the vector $(3, -4)$. The image $P'$ has coordinates $(2+3, 5-4)$, which is $(5, 1)$.
• A triangle with vertices $A(0,0)$, $B(1,3)$, and $C(4,1)$ is translated 2 units left and 5 units up. The new vertices are $A'(-2, 5)$, $B'(-1, 8)$, and $C'(2, 6)$.
• A translation moves point $Q(-1, 8)$ to $Q'(3, 5)$. The vector for this translation is $(3 - (-1), 5 - 8)$, which is $(4, -3)$.

Property

A reflection is a motion that leaves every point on a line $L$ (the line of reflection) fixed, and for a point $P$ not on $L$, with $P'$ its image, $L$ is the perpendicular bisector of the line segment $PP'$.

• A reflection preserves the lengths of line segments and the measures of angles.
• A reflection leaves every point on the line of reflection fixed and interchanges the two sides of that line.
• A reflection reverses orientation.

Examples

• The reflection of the point $P(4, 7)$ across the $x$-axis is the point $P'(4, -7)$. The $x$-coordinate stays the same, and the $y$-coordinate changes sign.
• The reflection of the point $Q(-2, 5)$ across the $y$-axis is the point $Q'(2, 5)$. The $y$-coordinate stays the same, and the $x$-coordinate changes sign.
• The reflection of the point $R(3, 8)$ across the line $y=x$ is the point $R'(8, 3)$. The coordinates are interchanged.