Glossary

9.1 Translations, reflections, and rotations

3 min read•july 22, 2024

Geometric transformations are like magic tricks for shapes. They let us move, flip, and spin figures around the coordinate plane. We'll learn about translations, reflections, and rotations - each with its own special rules.

These transformations are powerful tools for understanding how shapes behave. We'll see how to apply them to figures, calculate new coordinates, and explore how they affect a shape's size, position, and orientation.

Transformations in the Coordinate Plane

Types of geometric transformations

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  • Translations move every point of a figure the same distance in the same direction represented by a (a,b)(a, b), where aa is the horizontal shift and bb is the vertical shift
  • Reflections flip a figure over a line called the common lines include the xx-axis, yy-axis, and the lines y=xy = x and y=−xy = -x
  • Rotations turn a figure around a fixed point called the specified by an angle (in degrees) and a direction (clockwise or counterclockwise) common angles are 90°, 180°, and 270°

Applying transformations to figures

  • To translate a figure by a vector (a,b)(a, b):
    1. Add aa to the xx-coordinate of each point
    2. Add bb to the yy-coordinate of each point
  • To reflect a figure over a line:
    1. If reflecting over the xx-axis, negate the yy-coordinate of each point (yy-coordinate becomes its opposite)
    2. If reflecting over the yy-axis, negate the xx-coordinate of each point (xx-coordinate becomes its opposite)
    3. If reflecting over y=xy = x, swap the xx and yy coordinates of each point ((x,y)(x, y) becomes (y,x)(y, x))
    4. If reflecting over y=−xy = -x, swap the xx and yy coordinates and then negate both coordinates ((x,y)(x, y) becomes (−y,−x)(-y, -x))
  • To rotate a figure by an angle θ\theta counterclockwise around the origin:
    1. For each point (x,y)(x, y), the new coordinates are (xcos⁡θ−ysin⁡θ,xsin⁡θ+ycos⁡θ)(x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta)
    2. For clockwise rotations, use the opposite angle (−θ-\theta instead of θ\theta)

Coordinates after transformations

  • Apply the appropriate transformation rules to each point of the figure to determine the new coordinates
    • For translations, add the vector components to the respective coordinates (x+ax + a, y+by + b)
    • For reflections, negate or swap coordinates based on the line of (−x-x, −y-y, (y,x)(y, x), or (−y,−x)(-y, -x))
    • For rotations, use the rotation formulas with the given angle and direction ((xcos⁡θ−ysin⁡θ,xsin⁡θ+ycos⁡θ)(x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta) for counterclockwise, (xcos⁡(−θ)−ysin⁡(−θ),xsin⁡(−θ)+ycos⁡(−θ))(x \cos (-\theta) - y \sin (-\theta), x \sin (-\theta) + y \cos (-\theta)) for clockwise)

Effects of transformations on figures

  • Translations preserve the size, shape, and orientation of the figure but change its position (shift the figure without altering its appearance)
  • Reflections preserve the size and shape of the figure but change its orientation (create a mirror image across the line of reflection) and may change its position depending on the line of reflection
  • Rotations preserve the size and shape of the figure but change its orientation based on the angle of rotation (turn the figure around the center of rotation) and may change its position depending on the center of rotation

Key Terms to Review (18)

Angle of rotation theorem: The angle of rotation theorem states that the rotation of a figure around a point can be expressed as the angle through which the figure has been turned from its original position. This theorem is crucial for understanding how figures transform through rotations, allowing us to quantify the change in orientation and position of shapes on a plane. It connects deeply with concepts like symmetry, transformations, and coordinate geometry.
Center of rotation: The center of rotation is a fixed point in a plane around which a shape or object rotates. When an object rotates, every point of the object moves in a circular path around this center, maintaining the same distance from it. Understanding the center of rotation is essential for analyzing how shapes transform during rotations and for composing multiple transformations.
Congruence: Congruence refers to the property of geometric figures being identical in shape and size, allowing them to be superimposed onto one another without any gaps or overlaps. This concept is essential when comparing figures and helps in understanding their relationships, particularly in the study of angles, sides, and various transformations. Congruent figures maintain their properties through transformations like translations, reflections, and rotations, which are fundamental in geometry.
Line of reflection: A line of reflection is a specific line that acts as a mirror, where each point and its image are equidistant from this line. This concept is essential in understanding how reflections operate in geometry, allowing for the creation of congruent shapes and the preservation of distance and angles. The line serves as the axis across which a shape is flipped to create its reflected counterpart, maintaining the shape's dimensions and orientation relative to the line.
Line symmetry: Line symmetry, also known as reflectional symmetry, occurs when a figure can be divided into two identical halves that are mirror images of each other along a line. This concept is crucial in understanding how shapes can be transformed through reflections and how they relate to other symmetrical properties in both two and three dimensions.
M=1: In geometry, the notation m=1 typically signifies a slope of one, indicating that for every unit increase in the x-direction, there is a corresponding unit increase in the y-direction. This relationship is essential for understanding linear functions, where a slope of 1 represents a diagonal line that ascends at a 45-degree angle. This concept plays a crucial role in various transformations like translations, reflections, and rotations by providing insights into how shapes and lines interact with the coordinate plane.
R(90°): The term r(90°) refers to a rotation of 90 degrees about a specified point in a coordinate plane. This transformation involves turning a figure or point counterclockwise by 90 degrees around a center of rotation, which results in a new position for each point of the figure. Understanding r(90°) is crucial as it relates closely to other transformations like translations and reflections, which are fundamental concepts in geometry.
Reflection: Reflection is a transformation that flips a figure over a line, creating a mirror image of the original shape. This concept is crucial for understanding how shapes can be manipulated in space and relates to various geometrical operations like translations and rotations, symmetry in different dimensions, applications of vectors, and the composition of transformations.
Reflection rules: Reflection rules are guidelines that describe how a geometric figure is flipped over a specific line, known as the line of reflection, resulting in a mirror image of the original figure. These rules help in understanding how shapes and their coordinates change during the reflection transformation, making them crucial for grasping concepts related to symmetry and transformations in geometry.
Rotation Rules: Rotation rules refer to the guidelines used to determine how points in a geometric figure change position when the figure is rotated around a specific point, typically the origin. Understanding these rules helps in visualizing and performing rotations in the coordinate plane, which can be crucial for transforming shapes during various geometric operations.
Rotational Symmetry: Rotational symmetry is a characteristic of a shape that looks the same after being rotated by a certain angle around a central point. This concept is crucial in understanding how shapes and objects can be transformed and how they maintain their identity through various movements and orientations.
Symmetry: Symmetry refers to a balanced and proportional arrangement of elements in a figure or object, where one side mirrors or corresponds to the other side in some way. This concept can manifest in various forms, including reflectional, rotational, and translational symmetry, all of which play essential roles in understanding geometric properties and relationships. Symmetry is crucial for analyzing shapes, patterns, and designs in both theoretical and practical applications, making it a fundamental aspect of geometry.
T(x,y): The notation t(x,y) represents a translation transformation in the coordinate plane, moving a point or shape from its original position to a new location based on the specified horizontal and vertical distances. This transformation allows for the shifting of points in a way that preserves their orientation and size, making it fundamental in understanding how objects can be repositioned in a geometric space. Translations are defined by the movement along the x-axis and y-axis, resulting in new coordinates for any point.
Tiling Patterns: Tiling patterns are arrangements of shapes that cover a surface without any gaps or overlaps. These patterns can be created using transformations such as translations, reflections, and rotations, allowing for a diverse range of designs and configurations. The study of tiling patterns reveals important concepts about symmetry, geometry, and spatial relationships.
Transformation theorem: The transformation theorem is a fundamental principle that describes how geometric figures can be manipulated through various transformations, such as translations, reflections, and rotations, while maintaining their essential properties. This theorem ensures that the relationships between points, lines, and angles remain consistent before and after a transformation, demonstrating the concept of congruence and similarity in geometry.
Translation: Translation is a type of transformation that moves every point of a shape or object a certain distance in a specified direction, without changing its size, shape, or orientation. This concept is crucial for understanding how figures can be manipulated on a coordinate plane, and it connects to reflections, rotations, and various applications in geometry and vectors.
Translation rules: Translation rules are mathematical instructions that describe how to move a shape or figure from one location to another on a coordinate plane without altering its size, shape, or orientation. They play a crucial role in understanding transformations and help visualize how objects can be repositioned using specific guidelines based on their coordinates.
Vector: A vector is a mathematical object that has both magnitude and direction, often represented as an arrow in a coordinate system. Vectors are essential in understanding various geometric transformations and operations, as they provide a way to describe movements like translations, reflections, and rotations in space. The interplay of vectors with different mathematical operations allows for the analysis of geometric shapes and their properties in a clear and concise manner.
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