Question

the triangle lmn is a dilation of the triangle lmn. what is the scale factor of the dilation?

Explanation:

Step1: Find the length of a side in original triangle

Let's consider the vertical side \(LM\). The \(y -\)coordinate of \(L\) is \(2\) and of \(M\) is \(- 2\). Using the distance formula for vertical points (since \(x -\)coordinate is same for \(L\) and \(M\) i.e., \(x = 2\)), the length \(LM=\vert2-(- 2)\vert = 4\).

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

Consider the vertical side \(L'M'\). The \(y -\)coordinate of \(L'\) is \(8\) and of \(M'\) is \(-8\). Using the distance formula for vertical points (since \(x -\)coordinate is same for \(L'\) and \(M'\) i.e., \(x = 8\)), the length \(L'M'=\vert8-(-8)\vert = 16\).

Step3: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of side in dilated figure}}{\text{length of side in original figure}}\). Substituting the values, \(k = \frac{L'M'}{LM}=\frac{16}{4}=4\).

Answer:

The scale factor of the dilation is \(4\).