Question

the triangle ( abc ) is a dilation of the triangle ( abc ). 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: Identify corresponding side lengths

Take \(BB'\) and \(BC\). \(B=(0,3)\), \(B'=(0,4)\), \(C=(3,3)\). Length of \(BC = 3\) (from \(x = 0\) to \(x=3\) at \(y = 3\)), length of \(B'C'=6\) (from \(x = 0\) to \(x = 6\) at \(y = 4\)).

Step2: Calculate scale factor

Scale factor \(k=\frac{\text{length of side in }A'B'C'}{\text{length of corresponding side in }ABC}\). Using \(BC\) and \(B'C'\), \(k = \frac{4 - 4+ (6 - 0)}{3-0+ (4 - 3)}=\frac{6}{3}=2\) (Alternatively, using point \(A=(6,6)\) and \(A'=(8,8)\), the distance from the origin for \(A\): \(\sqrt{6^{2}+6^{2}}=\sqrt{72}\), for \(A'\): \(\sqrt{8^{2}+8^{2}}=\sqrt{128}\), scale factor \(k=\frac{\sqrt{128}}{\sqrt{72}}=\frac{8\sqrt{2}}{6\sqrt{2}}=\frac{4}{3}\) (wrong approach). Correct: Since dilation is uniform, using horizontal side \(BC = 3\) (from \(x = 0\) to \(x = 3\) for \(ABC\)) and \(B'C'=6\) (from \(x = 0\) to \(x = 6\) for \(A'B'C'\)). Scale factor \(k=\frac{6}{3}= 2\)

Answer:

\(2\)