Question

1. a figure is transformed by ((x, y) \to (x + 2, y - 3)) followed by ((x, y) \to (x, 3y)). does this sequence of transformations produce a pair of similar figures? explain your reasoning.

Explanation:

Step1: Analyze first transformation

The first transformation is \((x,y)\to(x + 2,y-3)\), which is a translation (a rigid transformation). Rigid transformations preserve the shape and size of the figure, so the figure after translation is congruent (and thus similar, with a scale factor of \(1\)) to the original.

Step2: Analyze second transformation

The second transformation is \((x,y)\to(x,3y)\), which is a vertical stretch with a scale factor of \(3\). Stretching (a similarity transformation) changes the size but preserves the shape (angles remain the same, and sides are scaled by a constant factor).

Step3: Combine transformations

A translation (rigid) followed by a vertical stretch (similarity) results in a figure that is similar to the original. Because rigid transformations preserve similarity, and then a similarity transformation (stretch) also preserves similarity (the composition of similarity transformations is a similarity transformation). The first transformation doesn't change the shape, and the second scales the \(y -\)coordinates by \(3\) (so all corresponding sides will be in proportion, and angles will be equal).

Answer:

Yes, this sequence of transformations produces a pair of similar figures. The first transformation \((x,y)\to(x + 2,y - 3)\) is a translation (rigid transformation, preserves shape and size). The second transformation \((x,y)\to(x,3y)\) is a vertical stretch (scale factor \(3\), preserves shape, scales size). The composition of a rigid transformation (which preserves similarity) and a similarity transformation (stretch) results in a figure similar to the original (corresponding angles are equal, corresponding sides are in proportion).