Question

triangle abc was dilated and translated to form similar triangle abc. what is the scale factor of the dilation?
\\( \frac { 1 } { 5 } \\)
\\( \frac { 2 } { 5 } \\)
\\( \frac { 5 } { 2 } \\)
\\( \frac { 5 } { 1 } \\)

Explanation:

Step1: Find the length of corresponding sides

Let's assume the length of side \(BC\) in \(\triangle ABC\). From the grid, \(BC = 2\) units.
The length of side \(B'C'\) in \(\triangle A'B'C'\). From the grid, \(B'C'= 5\) units.

Step2: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of side in image}}{\text{length of side in pre - image}}\).
Since \(\triangle ABC\) is the pre - image and \(\triangle A'B'C'\) is the image (after dilation and translation, translation does not affect the scale factor), \(k = \frac{B'C'}{BC}\).
Substitute \(B'C' = 5\) and \(BC=2\) into the formula: \(k=\frac{5}{2}\)

Answer:

\(\frac{5}{2}\) (corresponds to the third option)