Question

the triangle ( ijk ) is a dilation of the triangle ( ijk ). 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 in the original triangle

Let's consider the vertical side \(JK\). The \(y\) - coordinate of \(J\) is \(10\) and the \(y\) - coordinate of \(K\) is \(- 8\). Using the distance formula for a vertical line \(d=\vert y_2 - y_1\vert\), the length of \(JK=\vert10-(-8)\vert = 18\).

Step2: Find the length of the corresponding side in the dilated triangle

Consider the vertical side \(J'K'\). The \(y\) - coordinate of \(J'\) is \(5\) and the \(y\) - coordinate of \(K'\) is \(-4\). Using the distance formula for a vertical line \(d = \vert y_2 - y_1\vert\), the length of \(J'K'=\vert5-(-4)\vert=9\).

Step3: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of side in image}}{\text{length of side in pre - image}}\). So \(k=\frac{J'K'}{JK}=\frac{9}{18}=\frac{1}{2}\).

Answer:

$\frac{1}{2}$