Question

calculating a scale factor
quadrilateral qrst is dilated and translated to form similar figure qrst. what is the scale factor for the dilation?

Explanation:

Step1: Find the length of a side in the original figure

Let's consider side \(QR\). The length of \(QR\) can be found using the distance formula (or by counting units on the grid). If we assume the coordinates of \(Q\) and \(R\) (for simplicity, using the horizontal distance as the side is parallel to the \(x\) - axis). The length of \(QR = 4\) units (from \(x = 0\) to \(x = 4\) for the original figure).

Step2: Find the length of the corresponding side in the dilated figure

For the dilated figure, consider side \(Q'R'\). The length of \(Q'R'=2\) units (from \(x=-1\) to \(x = 1\) for the dilated figure).

Step3: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of side in dilated figure}}{\text{length of side in original figure}}\). So, \(k = \frac{Q'R'}{QR}=\frac{2}{4}=\frac{1}{2}\).

Answer:

$\frac{1}{2}$