Question

show your work here
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center (-2,8) and scale factor 3
center (-1,6) and scale factor 3
center (-2,8) and scale factor $\frac{7}{2}$
center (-1,6) and scale factor $\frac{7}{2}$
center (-2,8) and scale factor $\frac{1}{3}$
center (-1,6) and scale factor $\frac{1}{3}$

Explanation:

Step1: Recall dilation formula

For a dilation with center \((h,k)\) and scale - factor \(k\), the transformation of a point \((x,y)\) is given by \((x',y')=(h + k(x - h),k + k(y - k))\). We can also use the property that if we know a point on the original figure and its corresponding point on the dilated figure, we can find the center of dilation and scale - factor.

Step2: Assume a point on original and dilated figure

Let's assume a point on the original figure and its corresponding point on the dilated figure. We can use the fact that if the center of dilation is \((h,k)\), a point \((x_1,y_1)\) on the original figure and \((x_2,y_2)\) on the dilated figure, then the scale - factor \(s=\frac{x_2 - h}{x_1 - h}=\frac{y_2 - k}{y_1 - k}\).
Let's take a vertex of the original (black) and dilated (blue) polygons. Suppose a vertex of the original polygon is \((0,0)\) (for simplicity of calculation, if we can identify such a point). Let's assume the center of dilation is \((h,k)\).
If we consider the transformation of a point \((x,y)\) to \((x',y')\) under dilation with center \((h,k)\) and scale - factor \(s\), we have \(x'=h + s(x - h)\) and \(y'=k + s(y - k)\).
Let's try to find the center of dilation by looking at the relative position of the two polygons. We can observe that the center of dilation is a point such that the distances from the center to corresponding points on the two polygons are in the ratio of the scale - factor.
By visual inspection, if we consider the movement of the polygons, we can find the center of dilation. The center of dilation \((h,k)\) is a point around which the original polygon is expanded or contracted to get the dilated polygon.
If we assume a point \((x_1,y_1)\) on the original polygon and \((x_2,y_2)\) on the dilated polygon, and the center of dilation \((h,k)\), then \(s=\frac{x_2 - h}{x_1 - h}\).
Let's assume the center of dilation is \((- 2,8)\). If we take a point \((x_1,y_1)\) on the original polygon and \((x_2,y_2)\) on the dilated polygon and calculate the scale - factor \(s\) using the formula \(s=\frac{x_2+2}{x_1 + 2}=\frac{y_2 - 8}{y_1 - 8}\).
By looking at the size of the two polygons, we can see that the scale - factor \(s = 3\).
We can check this by taking a vertex of the original polygon, say \((0,0)\). If the center of dilation is \((-2,8)\) and scale - factor \(s = 3\), then the new point \((x',y')\) is given by \(x'=-2+3(0 + 2)=4\) and \(y'=8+3(0 - 8)=8-24=-16\).

Answer:

Center \((-2,8)\) and scale factor \(3\)