Practice Geometric Invariance in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

A property or measurement of a geometric figure that remains unchanged when a particular transformation is applied.

What stays exactly the same when you move, rotate, or flip a shape? Those unchanging things are invariants.

Showing a random 20 of 50 problems.

Example 1

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A right triangle with legs 33 and 44 is reflected, then translated. Find the length of its hypotenuse after the transformations.

Example 2

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Two figures are similar with ratio 2:32:3. If a length in the first measures 1010, find the corresponding length in the second.

Example 3

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A triangle has area 2020 and is reflected across a line, then dilated by scale factor 33. Compute the area of the final image.

Example 4

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A polygon has vertices A,B,C,DA,B,C,D in clockwise order. After a single reflection, are the image vertices in clockwise or counterclockwise order?

Example 5

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Is betweenness (one point lying between two others on a line) invariant under rigid motions?

Example 6

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A figure has 4-fold rotational symmetry about a point PP. Explain why any quantity preserved by rotation about PP is automatically the same for all four rotated copies.

Example 7

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Under rotation, a point is 5 units from the center. Name an invariant this illustrates.

Example 8

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A figure is reflected, rotated, then dilated by factor 22. If the original area is AA, find the final area in terms of AA.

Example 9

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Are angle measures invariant under rotation?

Example 10

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A property is invariant under translation and rotation but NOT reflection. Name such a property.

Example 11

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A triangle is reflected across the yy-axis and then rotated 90°90° counterclockwise about the origin. Which properties are invariant: (a) side lengths, (b) angle measures, (c) vertex orientation (CW vs CCW), (d) xx-coordinates of vertices?

Example 12

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Two figures are congruent. Which is true: they share all rigid-motion invariants, or all dilation invariants?

Example 13

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Which property is invariant under ALL four basic transformations (translation, rotation, reflection, dilation)?

Example 14

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How does the idea of invariance let you prove two figures are NOT congruent without measuring everything?

Example 15

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A rectangle is translated 5 units right and 3 units up. Name two properties that are invariant and one that is NOT invariant under this translation.

Example 16

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Is a figure's area invariant (unchanged) under a translation?

Example 17

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Name one quantity preserved by similarity but not by general affine transformations.

Example 18

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Under a glide reflection, list the rigid-motion invariants from {length, angle, orientation}.

Example 19

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True or False: the slope of a line is invariant under translation.

Example 20

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Is the number of sides of a polygon invariant under any rigid motion?
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