Question

the rectangle abcd is a dilation of the rectangle abcd. 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

Let's consider the vertical side. In rectangle \(ABCD\), if we take a vertical side (e.g., from \(D(0, - 3)\) to \(A(0,-6)\)), the length \(DA=\vert-6-(-3)\vert = 3\). In rectangle \(A'B'C'D'\), taking the corresponding vertical side (e.g., from \(D'(0, - 4)\) to \(A'(0,-8)\)), the length \(D'A'=\vert-8 - (-4)\vert=4\).

Step2: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of side in image}}{\text{length of side in pre - image}}\). So \(k = \frac{D'A'}{DA}\). Substituting the values we found: \(k=\frac{4}{3}\)

Answer:

\(\frac{4}{3}\)