High School: Geometry

High School: Geometry

Congruence HSG-CO.B.6

6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 

The words "rigid" and "motion" sound like complete opposites, don't they? Well, not in geometry. A rigid motion is a motion—or transformation, in geometric lingo—that preserves distance and lengths; in other words, every segment of the image is congruent to its preimage.

Students should already be comfortable with the ideas of translations, reflections, and rotations. They have to use rigid motions to not only transform figures and predict the effect it has on a specific figure, but also to determine when two figures are congruent.

Congruence between two geometric objects can be defined as a rigid motion on the coordinate plane that maps one object onto another. Students should be able to recite this this definition verbatim. Or, more importantly, they should actually understand it.

The key here is to differentiate between congruence and carrying one object onto another. While two reflected objects can't always be carried onto one another, they can still be congruent because all that separates them is reflection—a rigid motion. Make sense?

Once students fully understand how these topics fit together, they will be able to use transformations to not only find and prove properties of geometry, but to create patterns, including tessellations.

If your students are giving you a hard time with why these things are important, remind them that if they ever want to dabble in design, whether it's fashion, interior, or video game, they have to identify and create patterns. At the root of patterns is, you guessed it, geometry. That'll get them thinking about transformations and congruence the next time they watch Project Runway or play Call of Duty.

Drills

  1. There are three basic types of transformations: translation, rotation, and reflection. But there are four types of rigid transformations – translation, rotation, reflection, and one that combines two of these. Name and define the fourth one.

    Correct Answer:

    Glide reflection: a reflection over a line, then a translation in the same direction as that line

    Answer Explanation:

    It is a bit deceiving, given that the name does not entirely "reflect" its true meaning; however, to transform the image, it is first reflected over a line then translated in the same direction as that line of reflection. Although (A) sounds more logical for this, the terms are incorrect. Rotation isn't involved in the fourth rigid motion, which makes (B) and (C) incorrect.


  2. Which of the following sets of transformations would create an image that is not congruent to its original image?

    Correct Answer:

    A translation 4 units to the left followed by a dilation by 2

    Answer Explanation:

    Two images that are congruent should only be separated by rigid transformations (meaning translation, rotation, and reflection). All the transformations in (A), (B), and (D) are rigid, so the figures should be congruent to their original images. Dilating by 2, however, would create an image that is twice as large as the original one, and therefore not congruent.


  3. We know that a rigid motion is a transformation in which a figure's size and shape are not altered, although positioning can be altered. Given this, what is the definition of a non-rigid transformation?

    Correct Answer:

    Size, shape, and position can be altered

    Answer Explanation:

    As in rigid transformation, positioning can change after a non-rigid motion. In this case, so can shape and size. Answer (A) is incorrect, as this would mean non-rigid transformations are "more rigid." Answer (B) is incorrect because size and shape always go together in non-rigid transformations as the two items that can change. Since positioning can always be altered, (C) is also incorrect. Performing a non-rigid transformation is often called "skewing" or "distorting" the image, since non-rigid transformations can change pretty much anything.


  4. Rigid transformations are transformations that are not isometries. Is this true or false?

    Correct Answer:

    False, because the original figure and its transformed image are congruent in both isometries and rigid transformations

    Answer Explanation:

    The statement is false. As for the explanation, recall the definition of isometry: a transformation in which the original image is congruent to its transformed image. That makes (A) incorrect. Answer (B) is incorrect because even though the definition of isometry here is accurate, the statement should be false. Answer (D) is incorrect, because the original image and the transformed image must be congruent after rigid transformations as well. This is why (C) is correct.


  5. Each blade in a four-bladed windmill is identical to the other three. Which of the following is true about the rotation of the windmill's blades?

    Correct Answer:

    The transformation is rigid and the order of rotational symmetry is 4

    Answer Explanation:

    This is a rigid transformation since each blade of the windmill is the same size and shape, and that doesn't change as one is rotates "onto" the next. A non-rigid transformation would imply that each blade is different, which they aren't. Also, since the windmill has 4 blades, its order of rotational symmetry is 4. Our answer has to be (B).


  6. Is dilation a rigid or non-rigid transformation?

    Correct Answer:

    Dilation is non-rigid because the figure's size is not preserved

    Answer Explanation:

    A transformation is non-rigid when either the shape or the size of the image changes. In this case, although the shape does not change after transformation, the size will. That means compression is a non-rigid transformation. It keeps its shape, but still gets smaller. Isn't that what we'd all want?


  7. You're sitting across the table from a friend and you agree to share a triangular grilled cheese sandwich. You spin the plate toward your friend. Which transformations did you perform on the grilled cheese sandwich?

    Correct Answer:

    Translation and rotation

    Answer Explanation:

    First, it's good to notice that the grilled cheese only undergoes rigid transforms. Otherwise, it might stretch, grow, shrink, or mutate on the way. That'd make for an interesting, but significantly less edible grilled cheese. Since you're "spinning" the plate to your friend, we know that rotation is involved, but unless you and your friend are sitting in the same place, there's got to be some translation, too. That means (C) is the answer we're looking for.


  8. Which of the following is a rigid transformation?

    Correct Answer:

    Dribbling a basketball

    Answer Explanation:

    A rigid transformation, by definition, must meet the criterion that size and shape must be congruent, or stay the same, before and after the transformation. Cutting a piece of paper and inflating a balloon change both the size and shape of the object, so (B) and (C) aren't right. While the size of the laptop doesn't change, its shape certainly does. That means the only rigid transformation is the dribbling of a basketball.


  9. A triangle is dilated by 2, translated five units to the right, and then compressed by 2. Which of the following is true?

    Correct Answer:

    The transformations are non-rigid because the triangle does not stay the same size throughout

    Answer Explanation:

    Although the final triangle and the original triangle are congruent, the transformations performed aren't rigid because they don't maintain congruency throughout. Dilation increases the size of the image and compression decreases it. The triangle grows and shrinks by the same amount, but it doesn't change the fact that its size still changes.


  10. A square mirror has a horizontal scratch in the bottom right corner. If you reflect the mirror across a horizontal axis and then rotate it 90° counterclockwise, where will the scratch be and how will it be oriented?

    Correct Answer:

    Vertically in the top left corner

    Answer Explanation:

    If we reflect the mirror across a horizontal axis, the scratch will end up in the top right corner, still horizontal. Then, rotating the mirror 90° counterclockwise places the scratch in the top left corner. Rotating a horizontal line 90°, though, means it's changed from horizontal to vertical. That means our answer is (C). If you aren't sure, scratch a mirror and try it yourself.


More standards from High School: Geometry - Congruence

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