Question

select the correct answer from each drop - down menu.
pqrs and efgh are polygons that can be mapped onto each other using similarity transformations. the transformation that maps polygon pqrs onto polygon efgh is a followed by dilation with a scale factor of

Explanation:

Step1: Analyze the transformation type

To map polygon \(PQRS\) onto \(EFGH\), we first note that the orientation of the polygons has changed. A rotation is a transformation that changes the orientation of a figure. By observing the coordinate - plane, we can see that a rotation is required to align the polygons in a way that a dilation can then be applied.

Step2: Calculate the scale factor

Let's assume a pair of corresponding sides. For example, if we consider the vertical side of \(PQRS\) (say \(PQ\)) and the vertical side of \(EFGH\) (say \(EH\)).
Suppose the length of \(PQ\): If \(P=(4,12)\) and \(Q=(4,8)\), then \(PQ=\vert12 - 8\vert=4\).
If \(E=(14,6)\) and \(H=(14,10)\), then \(EH=\vert10 - 6\vert = 2\).
The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of side in image}}{\text{length of side in pre - image}}\). So \(k=\frac{EH}{PQ}=\frac{2}{4}=\frac{1}{2}\)

Answer:

rotation; \(\frac{1}{2}\)