Question

quadrilaterals pqrs and pqrs are shown below.
which series of transformations can be used to show that pqrs is similar to pqrs?
a. a 90° clockwise rotation about r followed by a dilation through r by scale factor $\frac{pq}{pq}$
b. a reflection across a vertical line passing through r followed by a dilation through r by a scale factor of $\frac{pq}{pq}$
c. a reflection across a vertical line passing through r followed by a dilation through r by a scale factor of $\frac{pq}{pq}$
d. a 90° clockwise rotation about r followed by a dilation through r by scale factor $\frac{pq}{pq}$

Explanation:

Step1: Analyze orientation

By observing the quadrilaterals, we can see that a reflection across a vertical line passing through \(R\) will align the orientation of \(PQRS\) with \(P'Q'R'S'\) as rotation by \(90^{\circ}\) will not give the correct orientation.

Step2: Determine scale - factor

For similar figures, the scale - factor of dilation should be the ratio of the corresponding side lengths of the image to the pre - image. The scale factor of dilation from \(PQRS\) to \(P'Q'R'S'\) is \(\frac{P'Q'}{PQ}\).

Answer:

C. a reflection across a vertical line passing through \(R\) followed by a dilation through \(R\) by a scale factor of \(\frac{P'Q'}{PQ}\)