Question

identify the center and scale factor of the dilation. the black shape is the pre - image and the blue shape is the image.

Explanation:

Step1: Find the center of dilation

The center of dilation is the point that remains fixed during the dilation. By observing the graph, we can see that the lines connecting corresponding points of the pre - image (black shape) and the image (blue shape) intersect at the point \((1,-1)\).

Step2: Calculate the scale factor

Let's take a pair of corresponding points. For example, if we consider a vertex of the pre - image and its corresponding vertex in the image.
Let's assume a point \(P(x_1,y_1)\) on the pre - image and \(P'(x_2,y_2)\) on the image. The scale factor \(k\) can be found using the formula \(k=\frac{\text{distance from center to image point}}{\text{distance from center to pre - image point}}\).
Another way is to use the ratio of the lengths of corresponding sides. If we count the number of units for a side of the pre - image and the corresponding side of the image.
Let's assume a side of the pre - image has a length \(l_1\) and the corresponding side of the image has a length \(l_2\).
By counting the units (using the grid), if we assume a side of the pre - image (black triangle) has a length of \(3\) units and the corresponding side of the image (blue triangle) has a length of \(6\) units. The scale factor \(k = 2\)

Answer:

Center: \((1,-1)\), Scale factor: \(2\)