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Quizzes > Mathematics

Which Transformations Keep Figures Congruent? Quiz

Quick, free quiz to test isometry vs dilation understanding. Instant results.

Editorial: Review CompletedCreated By: Jacob SmithUpdated Aug 28, 2025
Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Paper art representing a trivia quiz on geometric transformations for high school students.

This quiz helps you identify which transformations keep figures congruent and spot when a sequence preserves distance. Work through 20 quick questions with translations, rotations, reflections, and dilations, then deepen your skills with a dilation quiz, compare rules in similarity and congruence tests, or refresh basics with a triangle congruence quiz.

Which transformation shifts every point in the plane the same distance and direction?
Reflection
Translation
Dilation
Rotation
A translation moves every point by the same distance in a fixed direction, preserving the shape and size of the figure. There is no change in orientation or scale.
Which transformation produces a mirror image of a figure over a fixed line?
Dilation
Translation
Reflection
Rotation
Reflection creates a mirror-image of a figure about a designated line called the line of reflection. It reverses the orientation while preserving distances.
Which transformation rotates a figure about a fixed point by a specified angle?
Dilation
Translation
Rotation
Reflection
A rotation turns a figure around a fixed point by a given angle, keeping the distance from the center the same. The overall shape and size remain intact during the rotation.
Which transformation enlarges or reduces a figure proportionally?
Translation
Reflection
Dilation
Rotation
A dilation changes the size of a figure by a certain scale factor while maintaining its shape. The proportions of the figure remain constant, though its size increases or decreases.
Which transformation is represented by the matrix [[cosθ, -sinθ], [sinθ, cosθ]]?
Rotation
Translation
Dilation
Reflection
The provided matrix is the standard representation of a rotation in the plane about the origin by an angle θ. It maintains distances and angles, hence preserving the figure's congruence.
A point is first reflected across the line y = x and then reflected across the y-axis. Which transformation is equivalent to this combination?
Rotation of 90° counterclockwise about the origin
Reflection across the line y = x
Rotation of 90° clockwise about the origin
Reflection across the x-axis
Reflecting over y = x swaps the coordinates, and reflecting the result across the y-axis negates the new x-coordinate. This composition sends (x, y) to (-y, x), which is equivalent to a 90° counterclockwise rotation.
If a figure is dilated by a factor of 2 from the origin and then translated by the vector (3, -2), which statement best describes the overall effect?
The figure is rotated before being scaled.
The figure undergoes a reflection followed by translation.
The figure is shifted and then enlarged.
The figure is enlarged and then shifted to a new position.
The dilation enlarges the figure uniformly by a factor of 2, and subsequently the translation shifts every point by the same vector. The order in this case results in an enlarged figure that is repositioned.
What is the name for a transformation that involves a reflection across a line followed by a translation parallel to that line?
Rotation
Dilation
Glide Reflection
Shear
A glide reflection is a composite transformation combining a reflection with a translation along the direction of the reflecting line. It is the only isometry among these that results in no fixed point.
Which of the following properties remains unchanged under any rotation?
Position relative to the axes
Shape and size of the figure
Orientation of the figure
Angle with respect to the origin
Rotation is an isometry that preserves both the shape and size of a figure. Although the figure's position and orientation change, the distances and angles within the figure remain the same.
Reflecting a figure over two parallel lines results in which transformation?
Rotation
Translation
Glide Reflection
Dilation
When a figure is reflected over two parallel lines, the net effect is a translation. The distance of this translation is twice the distance between the two parallel lines.
What is the effect of performing a dilation with a scale factor less than 1 on a geometric figure?
The figure is enlarged.
The figure is reflected.
The figure is reduced in size.
The figure is rotated.
A dilation with a scale factor less than 1 compresses the figure, reducing all distances proportionally. The original shape is maintained though the figure becomes smaller in size.
When a figure is translated and then reflected, which of the following properties is always preserved?
Distances and angles (congruence)
Only distances, but not angles
Area and orientation
Only angle measures
Both translation and reflection are isometries, and as such they preserve distances and angles within the figure. The composite transformation guarantees that the figure remains congruent to its original shape.
A 180° rotation about the origin maps the point (x, y) to which coordinate?
(y, x)
(x, -y)
(-y, x)
(-x, -y)
Rotating a point 180° about the origin sends it to the point diametrically opposite, effectively negating both coordinates. This is a direct consequence of the properties of rotation in the plane.
If a figure is rotated 90° and then dilated by a factor of 3, which property is not preserved?
Orientation
Parallelism
Side lengths
Angles
While the rotation and subsequent dilation preserve angles and the overall shape (similarity), the dilation alters the side lengths of the figure. As a result, the figure is no longer congruent to its original form.
Which of the following is always preserved by an isometry such as translation, rotation, or reflection?
Size of the coordinate grid
Area of the figure
Distance between any two points
Measured angles relative to the origin
Isometric transformations preserve the distances between points, ensuring that shapes remain congruent. Although other attributes like orientation and position may change, the intrinsic dimensions of the figure remain constant.
A composite transformation rotates a figure 60° counterclockwise about the origin and then reflects it over the line y = √3x. Which single transformation is equivalent to this composition?
Glide reflection along the line y = (1/√3)x
Translation by (1, 1)
Rotation by 120° counterclockwise about the origin
Reflection across the line y = (1/√3)x
The rotation by 60° followed by a reflection over the line y = √3x produces a composite matrix that simplifies to [1/2, √3/2; √3/2, -1/2]. This matrix corresponds to a reflection about a line making a 30° angle with the positive x-axis, which is equivalent to reflecting across the line y = (1/√3)x.
What is the image of the point (4, 3) after a dilation by a factor of 1/2 about the origin followed by a 90° clockwise rotation about the origin?
(1.5, -2)
(-2, -1.5)
(2, -1.5)
(-1.5, 2)
The dilation first reduces (4, 3) to (2, 1.5). A 90° clockwise rotation then maps (2, 1.5) to (1.5, -2) by swapping coordinates and negating the new y-value, yielding the final image.
Which sequence of transformations will result in a figure being enlarged and then mirrored?
Dilation followed by reflection
Rotation followed by dilation
Reflection followed by rotation
Translation followed by reflection
First performing a dilation enlarges the figure, and following it with a reflection produces a mirror image of this enlarged figure. The process changes the scale as well as the orientation, resulting in a non-isometric transformation overall.
What are the coordinates of the image of the point (2, -1) after a 270° counterclockwise rotation about the origin followed by a translation of (-3, 4)?
(-2, 4)
(4, -2)
(2, -4)
(-4, 2)
A 270° counterclockwise rotation (equivalent to a 90° clockwise rotation) transforms (2, -1) to (-1, -2). The subsequent translation by (-3, 4) shifts it to (-4, 2), which is the final image of the point.
When an isometric transformation is followed by a dilation, which property of the figure is not preserved?
The collinearity of the points
The shape of the figure
The angles within the figure
The size and scale of the figure
While isometries preserve shape, angles, and distances, a dilation changes the scale of the figure. As a result, the size and side lengths are not preserved, even though the overall shape remains similar.
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Study Outcomes

  1. Analyze combinations of translations, rotations, reflections, and dilations in composite figures.
  2. Apply geometric transformations to accurately modify the position and orientation of shapes.
  3. Identify and describe the sequence of transformations used in solving transformation problems.
  4. Interpret visual and descriptive problem statements to determine the required transformation steps.
  5. Evaluate the effects of multiple transformations on a single geometric figure.

Transformation Combinations Cheat Sheet

  1. Understanding Geometric Transformations - Geometric transformations include translations, rotations, reflections, and dilations, each shaping how figures move or resize on a plane. Think of them as the ultimate shape-shifting toolkit that keeps your formulas in line!
  2. Translations - Translations slide a figure horizontally, vertically, or diagonally without changing its size or orientation. Imagine gliding a picture across the page: every point travels the same distance in the same direction like synchronized swimmers.
  3. Rotations - Rotations spin a figure around a fixed point (the center of rotation) by a specific angle, preserving size and shape. It's like turning a spinner on a pointer - how far you spin and which way you turn matters!
  4. Reflections - Reflections flip a figure over a line of reflection to create a mirror image, reversing orientation but keeping size and shape intact. Picture yourself striking a pose in front of a magic mirror that perfectly duplicates your form!
  5. Dilations - Dilations stretch or shrink a figure around a center point by a scale factor, altering size while preserving shape and proportionality. Think of zooming in or out on a photo - everything grows or shrinks together like magic.
  6. Rigid Transformations - Rigid transformations (translations, rotations, reflections) preserve both size and shape, so your pre-image and image are congruent twins. No distortion here - just perfect moves that maintain all angles and side lengths!
  7. Non‑Rigid Transformations - Non‑rigid transformations like dilations change a figure's size but not its shape, producing similar figures with proportional sides and equal angles. It's like resizing a photograph: the look stays the same, only the scale changes.
  8. Combining Transformations - When you perform multiple transformations in sequence - say a rotation followed by a translation - you can create complex movements. Remember, order matters: spinning then sliding gives a different result than sliding then spinning!
  9. Algebraic Representation - In coordinate geometry, you can write transformations as equations: translating by (a, b) sends point (x, y) to (x + a, y + b), while dilations use a scale factor k to map (x, y) to (kx, ky). This turns your graph paper into a transformation playground!
  10. Practice Makes Perfect - Working through diverse transformation problems builds your confidence and mastery, so tackle as many examples as you can. With consistent practice, you'll flip, spin, slide, and zoom through exams like a pro!
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