Question

the rectangle hijk is a dilation of the rectangle hijk. 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

Take side \(HK\). The \(x -\)coordinate of \(H\) is \(-8\) and the \(x -\)coordinate of \(K\) is \(4\). Using the distance formula for horizontal distance \(d=\vert x_2 - x_1\vert\), \(HK=\vert4-(-8)\vert = 12\).

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

Take side \(H'K'\). The \(x -\)coordinate of \(H'\) is \(-10\) and the \(x -\)coordinate of \(K'\) is \(5\). Using the distance formula for horizontal distance \(d = \vert x_2 - x_1\vert\), \(H'K'=\vert5-(-10)\vert=15\).

Step3: Calculate the scale factor

The scale factor \(k\) of a dilation is given by \(k=\frac{\text{length of side in image}}{\text{length of side in pre - image}}\). So \(k=\frac{15}{20}=\frac{3}{4}\) (We could also use vertical sides. For example, if we consider the vertical side \(IJ\) (from \(y=-4\) to \(y = 0\), length \(4\)) and \(I'J'\) (from \(y=-5\) to \(y = 0\), length \(5\)), scale factor \(k=\frac{5}{ \frac{20}{3}}=\frac{3}{4}\))

Answer:

$\frac{3}{4}$