Question

the smaller triangle is a pre-image of the bigger triangle. the center of dilation is (1,-1). what is the scale factor used to create the dilation?

Explanation:

Step1: Identify corresponding points

Let's take a vertex of the pre - image (smaller triangle) and its image (bigger triangle). Let's assume one vertex of the pre - image is \(A(-1, - 1)\) and its image is \(A'(9,-1)\) (from the graph, we can see the horizontal line \(y = - 1\) for these two points). The center of dilation is \(C(1,-1)\).

Step2: Calculate the distance from center to pre - image and image

The distance between the center of dilation \(C(1,-1)\) and the pre - image point \(A(-1,-1)\) is given by the distance formula \(d=\vert x_2 - x_1\vert\) (since \(y\) - coordinates are the same) \(d_{CA}=\vert-1 - 1\vert=\vert-2\vert = 2\).

The distance between the center of dilation \(C(1,-1)\) and the image point \(A'(9,-1)\) is \(d_{CA'}=\vert9 - 1\vert=\vert8\vert = 8\).

Step3: Calculate the scale factor

The scale factor \(k\) of a dilation centered at \(C\) is given by the ratio of the distance from the center to the image point to the distance from the center to the pre - image point. So \(k=\frac{d_{CA'}}{d_{CA}}=\frac{8}{2}=4\). Also, since the image is on the same line as the pre - image with respect to the center of dilation (and the image is larger), the scale factor is positive.

Answer:

\(4\) (the option corresponding to \(4\))