Question

the triangle $lmn$ is a dilation of the triangle $lmn$. 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 in both triangles

For triangle \(LMN\), let's consider the vertical side \(LM\). The \(y -\)coordinate of \(L\) is \(2\) and of \(M\) is \(- 2\). Using the distance formula for a vertical line \(d=\vert y_2 - y_1\vert\), \(LM=\vert2-(-2)\vert = 4\).
For triangle \(L'M'N'\), consider the vertical side \(L'M'\). The \(y -\)coordinate of \(L'\) is \(8\) and of \(M'\) is \(-8\). Using the distance formula for a vertical line \(d=\vert y_2 - y_1\vert\), \(L'M'=\vert8-(-8)\vert = 16\).

Step2: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of side in image}}{\text{length of corresponding side in pre - image}}\).
So, \(k=\frac{L'M'}{LM}\). Substituting the values \(L'M' = 16\) and \(LM = 4\), we get \(k=\frac{16}{4}\).

Answer:

\(4\)