Question

the dashed triangle is the image of the solid triangle. the center of dilation is (-4, -4). what is the scale factor used to create the dilation?

Explanation:

Step1: Select a point on the solid triangle and its corresponding point on the dashed triangle

Let's take the top - most point of the solid triangle which is (-4, - 4) and its corresponding point on the dashed triangle which is (-4, 5).

Step2: Calculate the distance from the center of dilation (-4,-4) to the point on the solid triangle and the dashed triangle

The distance from the center of dilation (-4,-4) to the point (-4, - 4) on the solid triangle is $d_1 = 0$. The distance from the center of dilation (-4,-4) to the point (-4, 5) on the dashed triangle is $d_2=\vert5 - (- 4)\vert=9$. Since the distance from the center of dilation to a point on the pre - image (solid triangle) is considered as the base distance and to the image (dashed triangle) is the new distance, and the center of dilation is one of the vertices of the solid triangle, we can also use another non - overlapping point. Let's take the right - most point of the solid triangle near the origin say (-2,-4) and its corresponding point on the dashed triangle (1,1).
The vector from the center of dilation (-4,-4) to the point (-2,-4) on the solid triangle: $\vec{v_1}=(-2-(-4),-4 - (-4))=(2,0)$
The vector from the center of dilation (-4,-4) to the point (1,1) on the dashed triangle: $\vec{v_2}=(1-(-4),1 - (-4))=(5,5)$
We use the ratio of the lengths of the vectors. The length of $\vec{v_1}=\sqrt{(2 - 0)^2+(0 - 0)^2}=2$ and the length of $\vec{v_2}=\sqrt{(5 - 0)^2+(5 - 0)^2}=\sqrt{25 + 25}=\sqrt{50}=5\sqrt{2}$. But a simpler way is to use the vertical or horizontal displacements.
Let's use the horizontal displacement of a non - center - of - dilation point. Consider a point on the solid triangle say (-2,-4) and its corresponding point on the dashed triangle (1,1). The horizontal distance from the center of dilation (-4,-4) to (-2,-4) is $x_1=\vert-2-(-4)\vert = 2$, and the horizontal distance from the center of dilation (-4,-4) to (1,1) is $x_2=\vert1-(-4)\vert=5$.
The scale factor $k$ is given by the ratio of the distance of a point on the image from the center of dilation to the distance of the corresponding point on the pre - image from the center of dilation.
The scale factor $k=\frac{3}{1}=3$. We can also check with vertical distances. The vertical distance from the center of dilation (-4,-4) to a point on the solid triangle (say (-4,-7)) and its corresponding point on the dashed triangle (-4,5). The distance from (-4,-4) to (-4,-7) is 3 and from (-4,-4) to (-4,5) is 9. The scale factor $k = 3$.

Answer:

3