Geometry Transformations: Reflections

Geometry Transformations

• Transformations in geometry are operations that manipulate shapes and figures on a plane.
• These operations preserve certain properties of the shapes, like size and shape (in some cases)
• Common transformations include translations, rotations, reflections, and dilations.

Reflections

• A reflection is a transformation that flips a shape over a line, called the line of reflection.
• The shape and its reflection are mirror images of each other across the line of reflection.
• Every point in the pre-image has a corresponding point in the image, and the distance of each point to the line of reflection is the same.
• For any point on the line of reflection, the point and its reflection coincide.
• The line of reflection acts as a mirror.
• The reflection of a point (x, y) across the x-axis is (x, −y).
• The reflection of a point (x, y) across the y-axis is (−x, y).
• The reflection of a point (x, y) across the line y = x is (y, x).
• Reflections preserve the shape and size of the figure.

Line of Reflection

• The line of reflection is the crucial element of the transformation in reflections.
• It is the axis about which the figure is flipped.
• Perpendicular distances from points on the pre-image to the line of reflection are equal to the corresponding perpendicular distances from the reflected points to the line of reflection.

Properties of Reflections

• Reflections preserve distance.
• Reflections preserve angle measures.
• Two lines reflected over a line of reflection remain parallel.
• A reflection in a line is an isometry, meaning the transformation does not change the size of the shape. The lengths of sides and measures of angles remain the same.
• The image of a shape under a reflection is congruent to the pre-image.

Examples of Reflections

• Reflecting a triangle across the x-axis.
• Reflecting a square across the line y = x
• Reflecting a point across a specified vertical or horizontal line.
• The reflection of a point (a, b) across the line y = -x is (-b, -a).

Types of Reflections

• Reflection across a line: a single line acts as the mirror
• Reflection across an axis (x-axis or y-axis): The reflection is over the coordinate axes.
• Reflections across other lines (e.g., y = 2x, 3y = 4x): The specified line acts as the mirror

Applications of Reflections

• Reflections are used in various fields like:
  • Architecture
  • Design
  • Computer graphics
  • Physics
  • Engineering
  • Geometry problems