Question

what translation and dilation will map △qrs onto its image? △qrs is translated ? ▲ units to the right, then dilated by a factor of 2 with a center of dilation at the origin.

Explanation:

Step1: Observe x - coordinates

To find the translation, look at the change in the x - coordinates of corresponding points. For example, if we consider point Q. Let the coordinates of Q be \((x_1,y_1)\) and of \(Q''\) be \((x_2,y_2)\). By counting the grid - squares, we can see that the x - coordinate of \(Q''\) is 8 units more than the x - coordinate of Q. So the translation is 8 units to the right.

Step2: Determine dilation factor

To find the dilation factor, we can compare the lengths of corresponding sides. For example, if we consider the horizontal side of \(\triangle QRS\) from Q to R and the corresponding horizontal side of \(\triangle Q''R''S''\) from \(Q''\) to \(R''\). The length of the side QR is 2 units and the length of the side \(Q''R''\) is 4 units. The dilation factor \(k=\frac{\text{length of side in image}}{\text{length of side in pre - image}}=\frac{4}{2} = 2\).

Answer:

\(\triangle QRS\) is translated 8 units to the right, then dilated by a factor of 2 with a center of dilation at the origin.