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Quiz: Transformations and Congruence

Test your understanding of geometric transformations, rigid motions, and congruence with these questions.

1. Which of the following are rigid motions?

  1. Translation, rotation, dilation
  2. Translation, rotation, reflection
  3. Rotation, reflection, dilation
  4. Only translation and rotation
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The correct answer is B. The three rigid motions are translation (slide), rotation (turn), and reflection (flip). These transformations preserve both distance and angle measure. Dilation is NOT a rigid motion because it changes the size of figures, even though it preserves shape and angle measure.

Concept Tested: Rigid Motion

See: Chapter 6: Rigid Motions

2. In transformation notation, if point A is transformed to create a new point, the new point is labeled:

  1. Point A
  2. Point A'
  3. Point B
  4. Point 1
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The correct answer is B. In transformation notation, the image of point A is labeled A' (pronounced "A prime"). The original figure is called the pre-image, and the result after transformation is called the image. This prime notation clearly distinguishes between original points and their transformed positions.

Concept Tested: Image and Pre-Image Notation

See: Chapter 6: Pre-Image and Image

3. A translation is also called:

  1. A slide
  2. A turn
  3. A flip
  4. A resize
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The correct answer is A. A translation is a "slide" that moves every point of a figure the same distance in the same direction. The figure doesn't rotate, flip, or change size—it simply slides to a new position. A rotation (B) is a turn, a reflection (C) is a flip, and a dilation (D) is a resize.

Concept Tested: Translation

See: Chapter 6: Translation

4. Why is a dilation NOT considered a rigid motion?

  1. It changes the orientation of the figure
  2. It changes the size of the figure
  3. It changes the angles of the figure
  4. It moves the figure to a different location
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The correct answer is B. A dilation changes the size of a figure by a scale factor, making it larger or smaller. Rigid motions must preserve distance (and therefore size), so dilation does not qualify. While dilations preserve angle measures and shape, the change in size disqualifies them from being rigid motions. Options A, C, and D describe properties that dilations don't necessarily change or that aren't the defining characteristic.

Concept Tested: Dilation vs Rigid Motion

See: Chapter 6: Rigid Motions and Congruence

5. What does CPCTC stand for?

  1. Congruent Parts Create Triangular Correspondences
  2. Corresponding Parts of Congruent Triangles are Congruent
  3. Coordinate Planes Create Triangle Congruence
  4. Congruent Parallelograms Can Transform Continuously
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The correct answer is B. CPCTC stands for "Corresponding Parts of Congruent Triangles are Congruent." This is a fundamental principle: once you prove two triangles are congruent, you can conclude that all their corresponding sides and angles are equal. This principle is frequently used in geometric proofs.

Concept Tested: CPCTC

See: Chapter 6: Congruent Figures

6. Two figures are congruent if and only if:

  1. They have the same area
  2. They have the same perimeter
  3. One can be mapped onto the other by a sequence of rigid motions
  4. They are both triangles
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The correct answer is C. The modern definition of congruence states that two figures are congruent if and only if one can be mapped onto the other through a sequence of rigid motions (translations, rotations, and/or reflections). This ensures they have the same size and shape. Having the same area (A) or perimeter (B) doesn't guarantee congruence. Option D is incorrect because congruence applies to all figures, not just triangles.

Concept Tested: Congruent Figures

See: Chapter 6: Congruent Figures

7. A point at (3, 5) is translated by T₍₋₂,₄₎. What are the coordinates of its image?

  1. (1, 9)
  2. (5, 1)
  3. (-2, 4)
  4. (6, 20)
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The correct answer is A. A translation T₍h,k₎ moves a point (x, y) to (x+h, y+k). Here, T₍₋₂,₄₎ means translate left 2 units and up 4 units. Starting at (3, 5): (3 + (-2), 5 + 4) = (1, 9). Option B reverses the operations. Option C gives the translation vector instead of the image. Option D incorrectly multiplies instead of adding.

Concept Tested: Translation Coordinate Rule

See: Chapter 6: Translation

8. When a point (4, -3) is reflected over the x-axis, its image is:

  1. (-4, -3)
  2. (4, 3)
  3. (-4, 3)
  4. (3, -4)
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The correct answer is B. Reflection over the x-axis changes the sign of the y-coordinate while keeping the x-coordinate the same. The rule is (x, y) → (x, -y). Applying this to (4, -3): (4, -(-3)) = (4, 3). Option A reflects over the y-axis. Option C reflects over both axes. Option D swaps coordinates (reflection over y=x).

Concept Tested: Reflection Over the x-axis

See: Chapter 6: Reflection

9. Two triangles have all corresponding sides equal in length and all corresponding angles equal in measure. Are they congruent?

  1. Yes, because equal corresponding parts means congruent
  2. No, they might be in different locations
  3. Only if they're the same color
  4. Only if they have the same orientation
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The correct answer is A. If two triangles have all corresponding sides equal and all corresponding angles equal, they are congruent by definition. While they may be in different locations or orientations, there exists a sequence of rigid motions that can map one onto the other, which is what defines congruence. Location (B) and color (C) are irrelevant to congruence. Orientation (D) can differ—reflections reverse orientation but preserve congruence.

Concept Tested: Congruence Definition

See: Chapter 6: Congruent Figures

10. You perform a 90° rotation followed by a translation. If you reverse the order (translation first, then rotation), will the final result be the same?

  1. Yes, the order doesn't matter for transformations
  2. No, the order of transformations matters; different orders give different results
  3. Yes, but only for rotations and translations
  4. It depends on the center of rotation and translation vector
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The correct answer is D. In general, the order of transformations matters—performing T₂ ∘ T₁ (T₁ first, then T₂) usually gives a different result than T₁ ∘ T₂. However, there are special cases where order doesn't matter. For a rotation and translation, whether order matters depends on the specific center of rotation and translation vector. If the rotation is around the origin and certain conditions are met, the results might be the same, but generally they differ. Option A is too broad, option B is too absolute, and option C is incorrect.

Concept Tested: Composition of Transformations

See: Chapter 6: Composition of Transformations

Score

Check your answers above. How did you do?

  • 9-10 correct: Excellent! You understand transformations and congruence very well.
  • 7-8 correct: Good work! Review the properties of rigid motions and transformation rules.
  • 5-6 correct: Study the three rigid motions and practice applying coordinate rules.
  • Below 5: Review the chapter carefully, especially translations, rotations, reflections, and the definition of congruence.