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Chapter 4: Problem 1

VOCABULARY A glide reflection is a combination of which two transformations?

Short Answer

Expert verified
A glide reflection is a combination of a Translation and a Reflection.

Step by step solution

01

Understanding the Definition

A glide reflection is a specific type of transformation in Geometry. It is a term used to describe the combination of two types of transformations. The first step to answer this question is understanding the definition of a glide reflection.
02

Applying the Definition

According to the definition, a glide reflection is a combination of a translation (or slide) and a reflection (or flip). So, the two transformations that combined to form a glide reflection are Translation and Reflection.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Translation
Translation in geometry is a type of transformation that 'slides' a figure from one position to another without rotating it. You can think of translation like moving a box in a straight line across the floor. There is no change in orientation or size, just a shift from point A to point B.

Key characteristics of translation include:
  • Uniform direction and distance: Every point of the figure shifts the same distance in the same direction.
  • No change in shape or size: The figure remains congruent to its original form.
  • Parallel movement: Lines that are parallel before the translation remain parallel after.
These properties help maintain the shape and dimensions of the figure post-translation, which corresponds with preserving congruence in geometric terms.
Reflection
Reflection is another fundamental geometric transformation. Imagine holding an image in front of a mirror: the mirror creates a 'flipped' version of things, exchanging left for right or up for down. In geometric terms, reflection involves flipping a figure over a line, called the line of reflection.

Critical attributes of reflection include:
  • The line of reflection acts as a mirror, equidistant from any point on the figure and its corresponding reflected point.
  • The reflected shape is congruent to its original, maintaining size and shape.
  • Orientation reversal: If a figure was clockwise, its reflection will be counterclockwise, and vice versa.
Understanding reflection helps when dealing with symmetry and creating mirror images within geometry.
Geometric Transformations
Geometric transformations cover all methods used to move or change a figure while preserving some properties, like size and shape. These transformations are divided into four main types: translation, rotation, reflection, and dilation.

Geometric transformations include:
  • Translation: Moves the figure without altering its appearance.
  • Rotation: Turns the figure around a fixed point.
  • Reflection: Flips the figure over a line, changing its orientation.
  • Dilation: Changes the size of the figure but keeps its shape proportional.
Each transformation has unique properties and effects on figures, allowing for a variety of combinations and sequences, such as the glide reflection achieved by combining translation and reflection concurrently. These transformations are foundational for understanding complex movements and patterns in geometry.

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Most popular questions from this chapter

During a presentation, a marketing representative uses a projector so everyone in the auditorium can view the advertisement. Is this projection a congruence transformation? Explain your reasoning.

THOUGHT PROVOKING Is the composition of a translation and a re? ection commutative? (In other words, do you obtain the same image regardless of the order in which you perform the transformations?) Justify your answer

Quadrilateral JKLM is mapped to quadrilateral J?K?L?M? using the dilation \((\mathrm{x}, \mathrm{y}) \rightarrow\left(\frac{3}{2} \mathrm{x}, \frac{3}{2} \mathrm{y}\right) .\) Then quadrilateral \(\mathrm{J}^{\prime} \mathrm{K}^{\prime} \mathrm{L}^{\prime} \mathrm{M}^{\prime}\) is mapped to quadrilateral \(\mathrm{J}^{\prime \prime} \mathrm{K}^{\prime \prime} \mathrm{L}^{\prime \prime} \mathrm{M}^{\prime \prime}\) using the translation \((\mathrm{x}, \mathrm{y}) \rightarrow(\mathrm{x}+3, \mathrm{y}-4) .\) The vertices of quadrilateral \(\mathrm{J}^{\prime} \mathrm{K}^{\prime} \mathrm{L}^{\prime} \mathrm{M}^{\prime}\) are \(\mathrm{J}^{\prime}(-12,0), \mathrm{K}^{\prime}(-12,18)\) \(\mathrm{L}^{\prime}(-6,18),\) and \(\mathrm{M}^{\prime}(-6,0) .\) Find the coordinates of the vertices of quadrilateral JKLM and quadrilateral \(\mathrm{J}^{\prime \prime} \mathrm{K}^{\prime \prime} \mathrm{L}^{\prime \prime} \mathrm{M}^{\prime \prime}\) . Are quadrilateral JKLM and quadrilateral \(\mathrm{J}^{\prime \prime} \mathrm{K}^{\prime \prime} \mathrm{L}^{\prime \prime} \mathrm{M}^{\prime \prime}\) similar? Explain.

\(\overline{\mathrm{PQ}},\) with endpoints \(\mathrm{P}(1,3)\) and \(\mathrm{Q}(3,2),\) is reflected in the \(y\)-axis. The image \(\overline{\mathrm{P}^{\prime} Q^{\prime}}\) is then reflected in the \(x\)-axis to produce the image \(\overline{\mathrm{P}^{\prime \prime} \mathrm{Q}^{\prime \prime}}.\) One classmate says that \(\overline{\mathrm{PQ}}\) is mapped to \(\overline{\mathrm{P}^{\prime \prime} \mathrm{Q}^{\prime \prime}}\) by the translation \((\mathrm{x}, \mathrm{y}) \rightarrow(\mathrm{x}-4, \mathrm{y}-5) .\) Another classmate says that \(\overline{\mathrm{PQ}}\) is mapped to \(\overline{\mathrm{P}^{\prime \prime} \mathrm{Q}^{\prime \prime}}\) by a \((2 \cdot 90)^{\circ},\) or \(180^{\circ}\), rotation about the origin. Which classmate is correct? Explain your reasoning.

Solve the equation. $$-2(8-y)=-6 y$$
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