Question

the rectangle ( nopq ) is a dilation of the rectangle ( nopq ). 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 a side of the original rectangle

The \(y -\)coordinate of \(N\) is \(8\) and the \(y -\)coordinate of \(Q\) is \(- 4\). The length \(NQ=\vert8-(-4)\vert = 12\).

Step2: Find the length of the corresponding side of the dilated rectangle

The \(y -\)coordinate of \(N'\) is \(6\) and the \(y -\)coordinate of \(Q'\) is \(- 2\). The length \(N'Q'=\vert6 - (-2)\vert=8\).

Step3: Calculate the scale factor

The scale factor \(k=\frac{\text{length of side of dilated rectangle}}{\text{length of side of original rectangle}}\). So \(k = \frac{8}{12}=\frac{2}{3}\) (Alternatively, using the horizontal side: \(NO\) has length from \(x = 0\) to \(x = 8\), length \(NO = 8\); \(N'O'\) has length from \(x = 0\) to \(x = 6\), length \(N'O'=6\). Scale factor \(k=\frac{6}{8}=\frac{3}{4}\))

Answer:

$\frac{3}{4}$