Rigid Transformations Reference Sheet
Rigid Transformations
Geometry Reference Sheet • Coordinate Rules, Isometries & Invariants
Translations Reflections Rotations
Rigid Motion (Isometry): Preserves distance (length) and angle measure.
Pre-image \(\cong\) Image (\(\Delta ABC \cong \Delta A'B'C'\))
1
Translation — "Slide"
Glides every point the same distance in the same direction
Preserves Orientation
Coordinate Rule
\((x, y) \to (x + h,\; y + k)\)
\(h > 0\): Right, \(h < 0\): Left • \(k > 0\): Up, \(k < 0\): Down
Example: Right 4, Down 3 \(\to (x+4, y-3)\)
Point \(A(1, 2) \to A'(1+4, 2-3) = A'(5, -1)\)
A B C \( \vec{v} \) A' B' C'
Key Properties
Points move along parallel lines of equal distance.
Corresponding sides are parallel: \(\overline{AB} \parallel \overline{A'B'}\).
2
Reflection — "Flip"
Reverses Orientation
Flips across a line of reflection (perpendicular bisector of each segment \(\overline{PP'}\)).
Across \(x\)-axis
\((x, y) \to (x, -y)\)
\(y\) changes sign
Across \(y\)-axis
\((x, y) \to (-x, y)\)
\(x\) changes sign
Across line \(y = x\)
\((x, y) \to (y, x)\)
Swap \(x\) and \(y\)
Across line \(y = -x\)
\((x, y) \to (-y, -x)\)
Swap & negate both
Line \(m\) (\(y\)-axis) A B C A' B' C'
Line of Reflection:
Points on line are invariant:
\(P(0, y) = P'(0, y)\)
Clockwise vertices become counterclockwise!
3
Rotation — "Turn"
Preserves Orientation
Turns around a center point by an angle. Standard: Counterclockwise (CCW).
\(90^\circ\) CCW (\(270^\circ\) CW)
\((x, y) \to (-y, x)\)
\(180^\circ\) Turn (half turn)
\((x, y) \to (-x, -y)\)
\(270^\circ\) CCW (\(90^\circ\) CW)
\((x, y) \to (y, -x)\)
O(0,0) \(90^\circ\) A B C A' B' C'
Center of Rotation:
The center point remains fixed:
\(O(0,0) \to O'(0,0)\)
Distance from center preserved: \(OA = OA'\)
Rigid Motion Properties & Invariants At-A-Glance
All rigid motions produce congruent figures
| Transformation | Informal | Side Lengths | Angle Measures | Orientation | Parallelism | Invariant Points (Fixed) |
|---|
| Translation | Slide | ✓ Preserved | ✓ Preserved | Same (Direct) | ✓ Preserved | None (unless \(h=k=0\)) |
| Reflection | Flip | ✓ Preserved | ✓ Preserved | Reversed (Opposite) | ✓ Preserved | Any point on line of reflection |
| Rotation | Turn | ✓ Preserved | ✓ Preserved | Same (Direct) | ✓ Preserved | Center of rotation only |
TIP 1 Reflections across an axis invert the opposite coordinate.
TIP 2 \(180^\circ\) rotation negates both signs: \((-x, -y)\).
TIP 3 Prime notation (\(A'\)) denotes the image after movement.
Geometry Reference • Rigid Transformations Isometry: from Greek isos (equal) + metron (measure) • Congruence Preserved
