Question

triangle stv was dilated with the origin as the center of dilation to form △ stv. what is the scale factor of the dilation? 1/3 2/3 3/2 3/1

Explanation:

Step1: Find the length of a corresponding side

Let's consider the vertical distance from \(S\) to \(T\) (or \(V\)) and from \(S'\) to \(T'\) (or \(V'\)).
Suppose \(S=(0,6)\) and \(S'=(0,2)\) (by counting the grid - squares).

Step2: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of a side in the image}}{\text{length of the corresponding side in the pre - image}}\).
If the length of the side in the pre - image (for the vertical side) \(l_1 = 6\) (from \(y = 6\) to \(y = 0\)) and the length of the side in the image \(l_2=2\) (from \(y = 2\) to \(y = 0\)). Then \(k=\frac{2}{6}=\frac{1}{3}\).

Another way:
Let's assume a non - vertical side. For example, if we consider the horizontal distance.
Suppose in \(\triangle STV\), the horizontal distance from \(T\) to the \(y\) - axis is \(3\) units (if \(T=(- 3,-6)\) approximately by counting grid - squares) and in \(\triangle S'T'V'\), the horizontal distance from \(T'\) to the \(y\) - axis is \(1\) unit (if \(T'=(-1,-2)\) approximately).
Using the scale factor formula \(k = \frac{\text{horizontal distance in image}}{\text{horizontal distance in pre - image}}\), we have \(k=\frac{1}{3}\).

Answer:

\(\frac{1}{3}\)