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: Select a corresponding side

Let's consider the horizontal side of the triangles. For example, the horizontal distance from \(O\) to \(N\) and from \(O'\) to \(N'\). The coordinates of \(O(- 5,1)\) and \(N(5, - 1)\), the length of \(ON=\sqrt{(5 + 5)^2+(-1 - 1)^2}=\sqrt{100 + 4}=\sqrt{104}\). The coordinates of \(O'(-10,2)\) and \(N'(10,-4)\), the length of \(O'N'=\sqrt{(10 + 10)^2+(-4 - 2)^2}=\sqrt{400+36}=\sqrt{436}\). Another way is to just consider the ratio of the horizontal displacements. The horizontal displacement from \(O\) to \(N\) is \(5-(-5)=10\), and from \(O'\) to \(N'\) is \(10-(-10) = 20\).

Step2: Calculate the scale - factor

The scale factor \(k\) of a dilation is the ratio of the length of a side in the dilated figure (\(N'O'P'\)) to the length of the corresponding side in the original figure (\(NOP\)). If we consider the horizontal displacements, \(k=\frac{\text{length of side in }N'O'P'}{\text{length of corresponding side in }NOP}\). Using the horizontal displacements we found above, \(k = 2\).

Answer:

2