Consider points in the plane as ordered pairs (x, y) and...

Consider points in the plane as ordered pairs $(x, y)$ and consider the function $f$ on the plane defined by $f(x, y)=(k x, k y)$, where $k$ is a non-zero constant. Show that $f$ is a similarity.

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Key Concepts

Dilation

Dilation is a type of similarity transformation that expands or contracts figures by a fixed scale factor relative to a central point, typically the origin. The function f(x, y) = (kx, ky) is an example of a dilation since it multiplies each coordinate by the same non-zero constant, thus scaling distances uniformly.

Euclidean Distance

Euclidean distance, defined as the square root of the sum of the squares of the differences between corresponding coordinates, is the standard measure of distance in the plane. Under the transformation f(x, y) = (kx, ky), every distance is multiplied by |k|, demonstrating that the scaling property of similarities holds with respect to Euclidean distance.

Similarity Transformation

A similarity transformation is a mapping in the plane that preserves angles and scales all distances by a constant factor. This means that, although the sizes of geometric figures change, their shapes remain the same and corresponding angles are congruent.

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