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 corresponding sides

Let's consider side \(NN'\). Point \(N=(0,1)\) and \(N'=(0,4)\). The length of \(NN' = 4 - 1=3\).
Point \(M=(0,0)\) and \(M'=(0, - 4)\). The length of \(MM'=|0-(-4)| = 4\) (but we can also use another pair. Let's use \(LN\) and \(L'N'\).
Coordinates of \(L=(-1,1)\), \(N=(0,1)\), so \(LN=\sqrt{(0 - (-1))^{2}+(1 - 1)^{2}}=1\).
Coordinates of \(L'=(-4,4)\), \(N'=(0,4)\), so \(L'N'=\sqrt{(0-(-4))^{2}+(4 - 4)^{2}}=4\).

Step2: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the ratio of the length of a side of the dilated figure (\(L'M'N'\)) to the length of the corresponding side of the original figure (\(LMN\)).
Using \(LN = 1\) (original side) and \(L'N'=4\) (dilated side), the scale factor \(k=\frac{L'N'}{LN}\)
Since \(L'N' = 4\) and \(LN = 1\), \(k = 4\)

Answer:

\(4\)