Question

the triangle ( vwx ) is a dilation of the triangle ( vwx ). 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: Determine the length of a side in the original triangle

Let's consider the vertical side \(VX\) in \(\triangle VWX\). The coordinates of \(V(-2,1)\) and \(X(-2,-3)\). Using the distance formula for a vertical line \(d = |y_2 - y_1|\), we have \(|1-(-3)|=4\).

Step2: Determine the length of the corresponding side in the dilated triangle

Consider the vertical side \(V'X'\) in \(\triangle V'W'X'\). The coordinates of \(V'(-6,3)\) and \(X'(-6,-9)\). Using the distance formula for a vertical line \(d = |y_2 - y_1|\), we have \(|3 - (-9)| = 12\).

Step3: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the ratio of the length of a side in the dilated figure to the length of the corresponding side in the original figure. So \(k=\frac{\text{Length of side in dilated triangle}}{\text{Length of side in original triangle}}\). Substituting the values from Step1 and Step2, we get \(k=\frac{12}{4}=3\).

Answer:

\(3\)