Reflection is a transformation process in which a figure is flipped over a line, known as the axis of reflection, to produce a mirror image. This section explores the concept of reflection in the coordinate plane, particularly focusing on reflections over the line $y = x$ and the line $x = 3$.

Reflection Over the Line y = x

Reflecting a point or shape over the line $y = x$ involves swapping the $x$ and $y$ coordinates of each point in the shape. This operation creates a mirror image across the line $y = x$.

Key Concepts

• Axis of reflection: The line $y = x$.
• Transformation rule: For a point $P(x, y)$, the reflected point $P'(y, x)$.

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Example 1: Reflecting a Single Point

Given a point $A(2, 3)$, find its reflection over the line $y = x$.

• Original coordinates: $A(x, y)$ = $(2, 3)$
• Reflected coordinates: $A'(y, x)$ = $(3, 2)$

Example 2: Reflecting a Shape

Reflect triangle ABC with vertices at $A(-2, 2)$, $B(-6, 5)$, and $C(-3, 6)$over the line $y = x$.

• Original vertices:
  • $A(-2, 2)$
  • $B(-6, 5)$
  • $C(-3,) 6$
• Reflected vertices:
  • $A'(2, -2)$
  • $B'(5, -6)$
  • $C'(6, -3)$

Reflection Over the Line x = 3

When reflecting a shape over the line $x = 3$, the formula for the reflected point $P'(6-x, y)$ is used. This formula adjusts the x-coordinate based on its original distance from the line $x = 3$, while the y-coordinate remains unchanged.

Key Concepts

• Axis of reflection: The line $x = 3$.
• Transformation rule: For any point $P(x, y)$, the reflected point is $P'(6-x, y)$.

Example 3: Reflecting a Single Point

Consider a point $B(5, 4)$. Find its reflection over $x = 3$.

• Original coordinates:$B(x, y)$= $(5, 4)$
• Apply reflection formula: $B'(6-x, y) = (1, 4)$

Example 4: Reflecting a Rectangle

Reflect a rectangle with vertices at $P(2, 1)$, $Q(2, 4)$, $R(6, 4)$, and $S(6, 1)$ over $x = 3$.

• Original vertices:
  • $P(2, 1)$, $Q(2, 4)$, $R(6, 4)$, $S(6, 1)$
• Reflected vertices using $P'(6-x, y)$:
  • $P'(4, 1)$, $Q'(4, 4)$, $R'(0, 4)$, $S'(0, 1)$