Question

the triangle nop is a dilation of the triangle nop. 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 the vertical side from \(O\) to \(P\) in \(\triangle NOP\) and from \(O'\) to \(P'\) in \(\triangle N'O'P'\).
The length of \(OP\) (using the \(y -\)coordinates of \(O(- 1,-1)\) and \(P(-1,3)\)): \(|3-(-1)| = 4\).
The length of \(O'P'\) (using the \(y -\)coordinates of \(O'(-1,-3)\) and \(P'(-1,9)\)): \(|9-(-3)| = 12\).

Step2: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of side in image}}{\text{length of corresponding side in pre - image}}\).
So, \(k=\frac{O'P'}{OP}\).
Substitute \(O'P' = 12\) and \(OP = 4\) into the formula: \(k=\frac{12}{4}=3\).

Answer:

\(3\)