Practice Geometric Invariance in Math

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

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.

Example 1

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

Example 2

hard
Under a dilation with scale factor k \neq 1 centred at the origin, a circle has centre (a, b) and radius r. Identify which properties of the circle are invariant and which are not.

Example 3

easy
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 4

hard
The cross-ratio of four collinear points is (AC \cdot BD)/(BC \cdot AD), an invariant under projective transformations. If A, B, C, D are at positions 0, 1, 3, 6 on a number line, compute the cross-ratio.
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