Question

the scale factor of this dilation is
to find the scale factor, compare the coordinates of point
and point
the scale
factor is
because

Explanation:

Step1: Identify corresponding points

Let's assume the smaller - triangle has vertices \(W(1,2)\), \(X(1,5)\), \(Y(3,2)\) and the larger - triangle has vertices \(R(1,11)\), \(S(1,4)\), \(T(3,4)\). We can compare the coordinates of a pair of corresponding points. Let's take point \(X(1,5)\) and point \(S(1,4)\). But a better - choice is to use a point that is not on the same vertical line. Let's use \(Y(3,2)\) and \(T(3,4)\).

Step2: Calculate the scale factor

The distance from the origin in the \(y\) - direction for point \(Y\) is \(y_Y = 2\), and for point \(T\) is \(y_T = 4\). The scale factor \(k\) for a dilation is found by the ratio of the coordinates of the image to the pre - image for corresponding points. If we consider the \(y\) - coordinates of corresponding points \(Y\) and \(T\) (since the \(x\) - coordinates are the same for this pair of points), the scale factor \(k=\frac{y_T}{y_Y}=\frac{4 - 0}{2 - 0}=2\). We can also check using the \(x\) - coordinates of other corresponding non - collinear points. For example, if we consider the horizontal distance from the \(y\) - axis for the pre - image and image points. The scale factor is \(2\) because the distance between corresponding points in the image is twice the distance between corresponding points in the pre - image.

Answer:

2