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Unit 4: Transformations, Congruence & Similarity: Shaping and Reshaping Space

The content explores the fascinating world of geometric transformations, including translations, reflections, rotations, and enlargements, explaining how these affect shapes' positions and sizes. Mastery of these transformations enables individuals to analyze visual patterns, apply geometric principles in real-world contexts, and communicate changes in orientations and sizes with mathematical precision. Furthermore, it delves into congruence and similarity, outlining the fundamental principles for identifying and proving the characteristics of shapes that maintain or alter their properties under various transformations.

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Sections

  • 4

    Transformations, Congruence & Similarity

    This section explores geometric transformations, including translations, reflections, rotations, and dilations, and discusses the concepts of congruence and similarity.

    Learning Practice

  • 4.1

    Mastering Transformations: Moving Shapes On The Coordinate Plane

    This section introduces geometric transformations such as translations, reflections, rotations, and enlargements, focusing on how shapes can be manipulated on the coordinate plane.

    Learning Practice

  • 4.1.1

    Introduction

    This section introduces students to transformations in geometry, highlighting the importance of understanding how shapes can be manipulated while preserving or altering their properties.

    Learning Practice

  • 4.1.2

    Key Terms

    This section defines essential geometric terms such as object, image, isometry, dilation, and the coordinate plane, providing a foundation for understanding transformations.

    Learning Practice

  • 4.1.2.1

    Object

    In this section, we explore the concept of geometric objects, transformations, and their invariant properties.

    Learning Practice

  • 4.1.2.2

    Image

    This section dives into transformations in geometry, focusing on how shapes can be moved, resized, or flipped while examining properties like congruence and similarity.

    Learning Practice

  • 4.1.2.3

    Isometry (Rigid Transformation)

    Isometries maintain the size and shape of geometric figures while altering their position in space.

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  • 4.1.2.4

    Dilation (Non-Rigid Transformation)

    Dilation is a transformation that changes the size of a figure while preserving its overall shape, allowing for the analysis of similar figures.

    Learning Practice

  • 4.1.2.5

    Coordinate Plane

    This section introduces the concept of the coordinate plane and various transformations, including translations, reflections, rotations, and dilations.

    Learning Practice

  • 4.1.3

    Translation (Slide)

    This section introduces translations as a fundamental transformation in geometry, describing how shapes can be slid across the coordinate plane without altering their size or shape.

    Learning Practice

  • 4.1.3.1

    Concept

    This section introduces transformations in geometry, focusing on how shapes can be manipulated while maintaining or altering their properties.

    Learning Practice

  • 4.1.3.2

    Coordinate Rule

    The Coordinate Rule explains the processes of translation in geometry, describing how a point or shape's coordinates change using a translation vector.

    Learning Practice

  • 4.1.3.3

    Invariant Properties

    This section discusses invariant properties within geometric transformations, focusing on how certain properties of shapes remain unchanged during transformations like translations, reflections, rotations, and dilations.

    Learning Practice

  • 4.1.3.4

    Practice Problems 1.1

    Practice Problems 1.1 focuses on practical exercises involving translations and understanding movement of geometric shapes on the coordinate plane.

    Learning Practice

  • 4.1.4

    Reflection (Flip)

    Reflections in geometry describe how shapes flip over a line, maintaining their size and shape while reversing their orientation.

    Learning Practice

  • 4.1.4.1

    Concept

    This section introduces transformations in geometry, focusing on how shapes can be moved or resized while maintaining certain properties.

    Learning Practice

  • 4.1.4.2

    Common Lines Of Reflection And Coordinate Rules

    This section explores the transformations of geometric figures through reflections, highlighting the coordinate rules associated with various lines of reflection.

    Learning Practice

  • 4.1.4.3

    Invariant Properties

    This section focuses on the invariant properties of geometric transformations, discussing how translations, reflections, rotations, and dilations affect size, shape, and orientation.

    Learning Practice

  • 4.1.4.4

    Example 4: Reflecting A Point Across Y = X

    This section illustrates the process of reflecting a point across the line y = x using coordinates.

    Learning Practice

  • 4.1.4.5

    Practice Problems 1.2

    This section explores reflection transformations, providing a series of practice problems to solidify understanding.

    Learning Practice

  • 4.1.5

    Rotation (Turn)

    Rotation is a transformation that turns a shape around a fixed point, altering its position while preserving its size and shape.

    Learning Practice

  • 4.1.5.1

    Concept

    This section introduces transformations in geometry, explaining how shapes can be moved, resized, and oriented while maintaining certain properties.

    Learning Practice

  • 4.1.5.2

    Common Rotations Around The Origin (0, 0) And Coordinate Rules

    This section covers the fundamental concepts of rotation transformations around the origin, including coordinate rules and invariant properties.

    Learning Practice

  • 4.1.5.3

    Invariant Properties

    Invariant properties refer to key characteristics that remain unchanged during geometric transformations.

    Learning Practice

  • 4.1.5.4

    Example 6: Rotating A Point 180 Degrees

    This section covers the concept of rotating a point 180 degrees around the origin, demonstrating the transformation using coordinate rules.

    Learning Practice

  • 4.1.5.5

    Practice Problems 1.3

    This section presents practice problems related to transformations—involving rotation, translation, and reflection—aiming to solidify understanding of these concepts.

    Learning Practice

  • 4.1.6

    Enlargement (Resizing / Dilation)

    Enlargement, or dilation, involves changing the size of a geometric shape while maintaining its overall shape and angles.

    Learning Practice

  • 4.1.6.1

    Concept

    This section explores the mathematical principles of transformations, congruence, and similarity of geometric shapes, emphasizing how these concepts help us analyze visual patterns.

    Learning Practice

  • 4.1.6.2

    Important Notes On Scale Factor (K)

    This section discusses the concept of scale factor in geometric transformations, particularly in dilation, highlighting how it affects the size and orientation of shapes.

    Learning Practice

  • 4.1.6.3

    Invariant Properties

    Invariant properties refer to the characteristics of geometric figures that remain unchanged after transformations such as translations, reflections, rotations, and enlargements.

    Learning Practice

  • 4.1.6.4

    Example 9: Enlarging A Triangle With A Fractional Scale Factor (Reduction)

    This section explores the concept of enlargement in geometry through dilations with fractional scale factors, illustrating how shapes can be resized and transformed.

    Learning Practice

  • 4.1.6.5

    Example 10: Enlarging With A Negative Scale Factor

    This section discusses enlargements of geometric figures using negative scale factors, highlighting how this transformation affects the position and orientation of shapes.

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  • 4.1.6.6

    Practice Problems 1.4

    This section focuses on practice problems related to enlargements in geometric transformations.

    Learning Practice

  • 4.1.7

    Connection To Statement Of Inquiry & Myp Focus (Transformations)

    Transformations in geometry reveal how shapes interact with space, with properties that are maintained or altered.

    Learning Practice

  • 5

    Combining Transformations: A Sequence Of Moves

    This section explores the concept of combining transformations in geometry, emphasizing the importance of the order in which these transformations are applied.

    Learning Practice

  • 5.1

    Introduction

    This section introduces the concepts of transformations, congruence, and similarity in geometry, emphasizing how shapes can be altered in position, size, or orientation.

    Learning Practice

  • 5.2

    Key Rule

    The Key Rule emphasizes the importance of performing transformations in a specific order, as each transformation affects the subsequent ones.

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  • 5.3

    Important Note

    This section emphasizes the significance of mastering transformations in geometry to analyze visual patterns effectively.

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  • 5.4

    Observation

    This section explores how understanding transformations, congruence, and similarity in geometry allows us to analyze and describe the changes shapes undergo in a systematic way.

    Learning Practice

  • 5.5

    Example 3: Three Transformations

    This section explores the concept of three specific transformations – enlargement, rotation, and reflection – and how they interact to produce new images.

    Learning Practice

  • 5.6

    Practice Problems 2.1

    This section emphasizes evaluating transformations through practice problems, focusing on rotation, translation, reflection, and their composite effects.

    Learning Practice

  • 5.7

    Connection To Statement Of Inquiry & Myp Focus (Combining Transformations)

    This section emphasizes the understanding of geometric transformations and their impact on shapes, allowing students to analyze visual patterns.

    Learning Practice

  • 6

    Congruence: Same Shape, Same Size

    This section explores the concept of congruence in geometry, focusing on the conditions for shapes to be considered congruent and introducing various congruence postulates for triangles.

    Learning Practice

  • 6.1

    Introduction

    This section introduces the fundamental concepts of transformations in geometry, including how shapes can be manipulated through translation, reflection, rotation, and dilation.

    Learning Practice

  • 6.2

    Key Terms

    This section introduces essential vocabulary related to transformations in geometry, including isometries and dilations.

    Learning Practice

  • 6.2.1

    Congruent

    This section explores the concepts of congruence, similarity, and transformations in geometry, emphasizing how shapes can be manipulated while maintaining their properties.

    Learning Practice

  • 6.2.2

    Corresponding Sides

    This section explains the essential principles of congruence in geometric shapes, focusing on corresponding sides and angles.

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  • 6.2.3

    Corresponding Angles

    This section explores the concept of corresponding angles in geometric transformations, particularly focusing on congruent shapes.

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  • 6.2.4

    Symbol For Congruence

    This section introduces the concept of congruence in geometry, focusing on how shapes can be identical in size and shape, represented by specific symbols.

    Learning Practice

  • 6.3

    Rule For Congruent Shapes

    This section explores congruence in shapes, defining congruent figures and the rules that determine congruence based on equal lengths and angles.

    Learning Practice

  • 6.3.1

    Sss (Side-Side-Side)

    The Side-Side-Side (SSS) congruence postulate states that if all three sides of one triangle are equal in length to the three corresponding sides of another triangle, the two triangles are congruent.

    Learning Practice

  • 6.3.2

    Sas (Side-Angle-Side)

    The SAS theorem states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.

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  • 6.3.3

    Asa (Angle-Side-Angle)

    The ASA criterion states that two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding angles and included side of another triangle.

    Learning Practice

  • 6.3.4

    Rhs (Right-Angle-Hypotenuse-Side)

    This section explores the Right-angle-Hypotenuse-Side (RHS) criterion for proving triangle congruence, focusing on right-angled triangles.

    Learning Practice

  • 6.4

    Important Non-Rule

    The SSA congruence condition does not guarantee triangle congruence, leading to potentially ambiguous results.

    Learning Practice

  • 6.5

    Practice Problems 3.1

    This section presents practice problems focused on triangle congruence, encouraging students to determine congruency using the SSS, SAS, ASA, and RHS postulates.

    Learning Practice

  • 6.6

    Connection To Statement Of Inquiry & Myp Focus (Congruence)

    This section discusses how transformations in geometry relate to congruence and similarity, crucial for understanding visual patterns.

    Learning Practice

  • 7

    Similarity: Same Shape, Different Size

    This section explores the concepts of similarity in geometry, highlighting how shapes can maintain the same proportions while differing in size.

    Learning Practice

  • 7.1

    Introduction

    This section introduces the concept of transformations in geometry, focusing on how shapes can be altered while maintaining or changing their properties.

    Learning Practice

  • 7.2

    Key Terms

    This section defines essential geometric terms associated with transformations, providing a foundational understanding of how shapes can be manipulated in space.

    Learning Practice

  • 7.3

    Rule For Similar Shapes

    This section focuses on the characteristics of similar shapes, emphasizing the importance of equal angles and proportional corresponding sides.

    Learning Practice

  • 7.1

    Identifying Similar Shapes And Calculating Scale Factor

    This section explores how to identify similar shapes and calculate the scale factor between them.

    Learning Practice

  • 7.1.2

    Practice Problems 4.1

    This section emphasizes understanding and identifying similar figures through practice problems that involve scale factors and proportional reasoning.

    Learning Practice

  • 7.2

    Using Similarity To Find Unknown Lengths In 2d Shapes

    This section introduces the concept of similarity in 2D shapes and demonstrates how to use scale factors and proportions to determine unknown lengths.

    Learning Practice

  • 7.2.1

    Method 1: Using The Scale Factor

    This section focuses on how to determine similar shapes by using the scale factor to compare their dimensions.

    Learning Practice

  • 7.2.2

    Method 2: Using Proportions (Ratios)

    This section outlines the method of using proportions to solve problems involving similar figures, emphasizing the significance of ratios and scale factors.

    Learning Practice

  • 7.2.3

    Example 4: Similar Triangles Often Appear Nested Or Overlapping

    This section discusses the nature of similar triangles, particularly when they are nested or overlapping, emphasizing how their properties maintain consistent angles and proportional side lengths.

    Learning Practice

  • 7.2.4

    Practice Problems 4.2

    This section presents practice problems designed to reinforce the concepts of transformations, congruence, and similarity in geometry.

    Learning Practice

  • 7.3

    Connection To Statement Of Inquiry & Myp Focus (Similarity)

    This section explores the relationships between congruence and similarity through geometric transformations and their properties within systems.

    Learning Practice

References

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Class Notes

Memorization

What we have learnt

  • Transformations change a ge...
  • Congruent shapes are exact ...
  • Understanding transformatio...

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