Question

is efgh similar to pqrs? justify your answer. yes, because a dilation by a scale factor of \\( \frac { 1 } { 2 } \\) centered at the origin maps efgh onto pqrs. yes, because a dilation by a scale factor of \\( \frac { 1 } { 2 } \\) centered at point e maps efgh onto pqrs. no, because \\( \angle g \\) and \\( \angle r \\) do not have the same measure.

Explanation:

Step1: Recall the properties of similar figures

Similar figures can be mapped to each other by a sequence of rigid motions and dilations.

Step2: Analyze the dilation center

If we consider a dilation centered at the origin, we can check the side - length ratios. If we assume the coordinates of the vertices (for example, if \(EFGH\) has side - lengths that are twice the side - lengths of \(PQRS\) after a dilation centered at the origin with a scale factor of \(\frac{1}{2}\), the side - length ratios will be consistent. A dilation centered at a non - origin point (like point \(E\)) will not map \(EFGH\) onto \(PQRS\) in a way that preserves the similarity ratios for all sides. Also, for polygons, if they are similar, all corresponding angles are equal. Since dilation is a similarity transformation (preserves angle measures), we don't have an issue with angle measures (so the option about angles is incorrect).

Answer:

Yes, because a dilation by a scale factor of \(\frac{1}{2}\) centered at the origin maps \(EFGH\) onto \(PQRS\).