Question

the triangle ( efg ) is a dilation of the triangle ( efg ). 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 side in the original triangle and the corresponding side in the dilated triangle

Let's consider side \(FG\). The coordinates of \(F\) is \((-1,1)\) and \(G\) is \((1,1)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), for \(FG\) with \(x_1=-1,y_1 = 1,x_2=1,y_2 = 1\), we have \(FG=\sqrt{(1-(-1))^2+(1 - 1)^2}=\sqrt{(2)^2+0^2}=2\).
For the corresponding side \(F'G'\), the coordinates of \(F'\) is \((-4,1)\) and \(G'\) is \((4,1)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), with \(x_1=-4,y_1 = 1,x_2=4,y_2 = 1\), we have \(F'G'=\sqrt{(4-(-4))^2+(1 - 1)^2}=\sqrt{(8)^2+0^2}=8\).

Step2: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of side in dilated figure}}{\text{length of corresponding side in original figure}}\).
Substituting the values we found: \(k=\frac{F'G'}{FG}\). Since \(F'G' = 8\) and \(FG=2\), then \(k=\frac{8}{2}=4\).

Answer:

\(4\)