Coordinate Moves: Reflecting Polygons Across the Coordinate Axes

Principles and Definitions of Line Reflections

• A reflection is a rigid transformation (an isometry) that maps every point of a geometric figure across a specified line of reflection (also called the mirror line).
• Core Geometric Definition of Reflection Across a Line: To reflect a point over a line, the mapped image point must lie at the exact same perpendicular distance from the line of reflection, but in the opposite direction from the line.
• Operational Procedure on a Coordinate Plane:
  • Identify the line of reflection (such as the $x$-axis, given by the line equation $y = 0$, or the $y$-axis, given by the line equation $x = 0$).
  • For each individual point or vertex, calculate its perpendicular distance from the reflection line.
  • Plot the corresponding image point on the opposite side of the line along the perpendicular path, preserving that exact same distance.
• Algebraic Rules for Coordinate Axes Reflections:
  • Reflection across the $x$-axis: When reflecting a point $(x, y)$ over the horizontal axis ($x$-axis), the horizontal coordinate remains unchanged while the sign of the vertical coordinate is inverted:     $(x, y) \rightarrow (x, -y)$
  • Reflection across the $y$-axis: When reflecting a point $(x, y)$ over the vertical axis ($y$-axis), the vertical coordinate remains unchanged while the sign of the horizontal coordinate is inverted:     $(x, y) \rightarrow (-x, y)$

Coordinate Analysis and Reflection of Quadrilateral QRST Across the x-Axis

• Quadrilateral $QRST$ is located initially in Quadrant IV, where horizontal coordinates ($x$) are positive and vertical coordinates ($y$) are negative.
• The Cartesian coordinate plane displays horizontal axis intervals from $-10$ to $10$ and vertical axis intervals from $-8$ to $10$, with labeled grid intervals marked every $2$ units.
• Vertex-by-Vertex Transformation over the $x$-axis ($y = 0$):
  • Vertex $Q$:
  • Original coordinate: $(3, -6)$
  • Distance from the $x$-axis: $|-6| = 6\text{ units}$ in the negative vertical direction (downward).
  • Transformed image point $Q'$: Located $6\text{ units}$ in the positive vertical direction (upward) along the line $x = 3$.
  • Reflected coordinate: $Q'(3, 6)$
  • Vertex $R$:
  • Original coordinate: $(3, -4)$
  • Distance from the $x$-axis: $|-4| = 4\text{ units}$ downward.
  • Transformed image point $R'$: Located $4\text{ units}$ upward along the vertical line $x = 3$.
  • Reflected coordinate: $R'(3, 4)$
  • Vertex $S$:
  • Original coordinate: $(7, -5)$
  • Distance from the $x$-axis: $|-5| = 5\text{ units}$ downward.
  • Transformed image point $S'$: Located $5\text{ units}$ upward along the vertical line $x = 7$.
  • Reflected coordinate: $S'(7, 5)$
  • Vertex $T$:
  • Original coordinate: $(8, -6)$
  • Distance from the $x$-axis: $|-6| = 6\text{ units}$ downward.
  • Transformed image point $T'$: Located $6\text{ units}$ upward along the vertical line $x = 8$.
  • Reflected coordinate: $T'(8, 6)$
• Summary Table of Initial Reflection:
  • Original Point $Q(3, -6)$ maps to image point $Q'(3, 6)$
  • Original Point $R(3, -4)$ maps to image point $R'(3, 4)$
  • Original Point $S(7, -5)$ maps to image point $S'(7, 5)$
  • Original Point $T(8, -6)$ maps to image point $T'(8, 6)$
• The resulting figure, quadrilateral $Q'R'S'T'$, lies completely within Quadrant I, retaining identical edge lengths and interior angle measurements as quadrilateral $QRST$ because reflection is a distance-preserving isometry.

Successive Reflection: Reflecting Image Q'R'S'T' Across the y-Axis

• The multi-step transformation sequence requires reflecting the newly produced image, quadrilateral $Q'R'S'T'$, across the vertical axis ($y$-axis, $x = 0$).
• Applying the horizontal reflection rule $(x, y) \rightarrow (-x, y)$ to the vertices of $Q'R'S'T'$ maps the figure from Quadrant I to Quadrant II:
  • Vertex $Q'$ to $Q''$:
  • Image coordinate: $Q'(3, 6)$
  • Distance from the $y$-axis: $|3| = 3\text{ units}$ in the positive horizontal direction (to the right).
  • Final coordinate: $Q''(-3, 6)$, located $3\text{ units}$ to the left of the $y$-axis along $y = 6$.
  • Vertex $R'$ to $R''$:
  • Image coordinate: $R'(3, 4)$
  • Distance from the $y$-axis: $|3| = 3\text{ units}$ to the right.
  • Final coordinate: $R''(-3, 4)$, located $3\text{ units}$ to the left along $y = 4$.
  • Vertex $S'$ to $S''$:
  • Image coordinate: $S'(7, 5)$
  • Distance from the $y$-axis: $|7| = 7\text{ units}$ to the right.
  • Final coordinate: $S''(-7, 5)$, located $7\text{ units}$ to the left along $y = 5$.
  • Vertex $T'$ to $T''$:
  • Image coordinate: $T'(8, 6)$
  • Distance from the $y$-axis: $|8| = 8\text{ units}$ to the right.
  • Final coordinate: $T''(-8, 6)$, located $8\text{ units}$ to the left along $y = 6$.

Composite Transformations and Geometric Equivalence

• Composition of Perpendicular Reflections:
  • Let $r_x$ represent reflection over the $x$-axis and $r_y$ represent reflection over the $y$-axis.
  • The composite operation $(r_y \circ r_x)$ maps any point $(x, y)$ as follows:     $(x, y) \xrightarrow{r_x} (x, -y) \xrightarrow{r_y} (-x, -y)$
• Equivalence to Half-Turn (Rotation):
  • Any sequence of two reflections across perpendicular lines intersecting at an origin $(0, 0)$ is mathematically equivalent to a single rotation of $180^\circ$ (either clockwise or counterclockwise) around that point of intersection:     $R_{(0,0), 180^\circ}(x, y) = (-x, -y)$
  • Direct verification of vertices from initial figure $QRST$ to final figure $Q''R''S''T''$:
  • $Q(3, -6) \rightarrow Q''(-3, 6)$
  • $R(3, -4) \rightarrow R''(-3, 4)$
  • $S(7, -5) \rightarrow S''(-7, 5)$
  • $T(8, -6) \rightarrow T''(-8, 6)$
• Orientation Properties:
  • A single reflection reverses the orientation (chirality) of a polygon from clockwise to counterclockwise (or vice versa).
  • Two successive reflections restore the original orientation of the polygon, consistent with the properties of pure rotations.