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 the horizontal side \(QR\) in the original quadrilateral \(QRST\). If we assume the grid - square has a side - length of 1 unit. The length of \(QR\) can be found using the distance formula for horizontal points (since \(y\) - coordinates are the same). If \(Q\) is at \((-1,0)\) and \(R\) is at \((5,0)\), then the length \(QR=\vert5 - (- 1)\vert=6\) units.

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

Consider the corresponding side \(Q'R'\) in the dilated quadrilateral \(Q'R'S'T'\). If \(Q'\) is at \((-1,2)\) and \(R'\) is at \((1,2)\), then the length \(Q'R'=\vert1-(-1)\vert = 2\) units.

Step3: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of side in dilated figure}}{\text{length of corresponding side in original figure}}\). So, \(k=\frac{Q'R'}{QR}\). Substituting the values we found: \(k = \frac{2}{6}=\frac{1}{3}\)

Answer:

\(\frac{1}{3}\)