Question

the triangle wxy is a dilation of the triangle wxy. what is the scale factor of the dilation? simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

Explanation:

Step1: Find the length of a corresponding side

Let's consider the vertical side of the triangles. For example, the vertical side from \(X\) to \(W\) in \(\triangle WXY\) and the vertical side from \(X'\) to \(W'\) in \(\triangle W'X'Y'\).
The length of \(XW\): \(y\) - coordinate of \(W\) is \(- 6\), \(y\) - coordinate of \(X\) is \(-2\). So \(| - 6-(-2)|=4\).
The length of \(X'W'\): \(y\) - coordinate of \(W'\) is \(3\), \(y\) - coordinate of \(X'\) is \(1\). So \(|3 - 1| = 2\).

Step2: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of a side in the image}}{\text{length of the corresponding side in the pre - image}}\).
Here, \(k=\frac{X'W'}{XW}\). Substituting the values we found: \(k = \frac{2}{4}=\frac{1}{2}\)

Answer:

\(\frac{1}{2}\)