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 corresponding side in both rectangles

For rectangle \(HIJK\), let's consider the side \(HK\). The \(x -\)coordinate of \(H\) is \(- 8\) and the \(x -\)coordinate of \(K\) is \(4\). Using the distance formula for a horizontal line \(d=\vert x_2 - x_1\vert\), \(HK=\vert4-(-8)\vert = 12\).
For rectangle \(H'I'J'K'\), consider the side \(H'K'\). The \(x -\)coordinate of \(H'\) is \(-10\) and the \(x -\)coordinate of \(K'\) is \(5\). Using the distance formula for a horizontal line \(d = \vert x_2-x_1\vert\), \(H'K'=\vert5 - (- 10)\vert=15\).

Step2: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of side in image}}{\text{length of side in pre - image}}\).
\(k=\frac{HK}{H'K'}\) (since \(HIJK\) is the pre - image and \(H'I'J'K'\) is the image). Substituting the values: \(k=\frac{12}{15}\).
Simplify the fraction \(\frac{12}{15}\) by dividing both the numerator and the denominator by their greatest common divisor, which is \(3\). \(\frac{12\div3}{15\div3}=\frac{4}{5}\).

Answer:

\(\frac{4}{5}\)