DISCOVER ■ PROVE: Geometric Invariants Do you expect that the distance between two points is invariant under rotation? Prove your answer by comparing the distance and where and are the images of and under a rotation of axes.
Whether the distance between two points is invariant or not under rotation and the proof for it.
The coordinates and are the original points.
The coordinates and are the images of and under the rotation of axes.
A quantity is invariant under rotation if it does not change when the axes is rotated.
The formulas for the rotation of axes is given as,
Here, and are the points for the image and and are the original points.
The distance formula is given as,
The coordinates are given as,
The coordinates for the images are given as,
Substitute the values of and in the coordinates for the image as,
Compute the distance between and using the distance formula as,
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