Question

marty sketches triangle def on the coordinate plane. he then sketches triangle def on the same plane. what is the scale factor going from def to def?

Explanation:

Step1: Recall the formula for scale factor

The scale factor \(k\) for a dilation from a pre - image (triangle \(DEF\)) to an image (triangle \(D'E'F'\)) with a center of dilation at the origin \((0,0)\) is given by \(k=\frac{x_{image}}{x_{pre - image}}\) (or \(\frac{y_{image}}{y_{pre - image}}\)) for corresponding points.

Step2: Choose a pair of corresponding points

Let's use point \(F(9,6)\) (pre - image) and \(F'(6,4)\) (image).
For the \(x\) - coordinates: \(k=\frac{x_{F'}}{x_{F}}\), substituting \(x_{F'}=6\) and \(x_{F}=9\), we get \(k = \frac{6}{9}=\frac{2}{3}\).
For the \(y\) - coordinates: \(k=\frac{y_{F'}}{y_{F}}\), substituting \(y_{F'}=4\) and \(y_{F}=6\), we get \(k=\frac{4}{6}=\frac{2}{3}\).
We can also check with point \(E(6, - 3)\) and \(E'(4,-2)\).
For the \(x\) - coordinates: \(k=\frac{x_{E'}}{x_{E}}=\frac{4}{6}=\frac{2}{3}\).
For the \(y\) - coordinates: \(k=\frac{y_{E'}}{y_{E}}=\frac{-2}{-3}=\frac{2}{3}\).

Answer:

The scale factor is \(\frac{2}{3}\)