Question

△rst is dilated with the rule dr,1/3 (x,y), where the center of dilation is t(3, -2).
the distance between the x - coordinates of r and t is ▢.
the distance between the y - coordinates of r and t is ▢.
r is ▢ from t, so the coordinates of r are ▢.

Explanation:

Step1: Find the distance between x - coordinates of R and T

The x - coordinate of R is \(4\), and the x - coordinate of T is \(3\). The distance is \(|4 - 3|=1\). Multiply by the scale factor \(\frac{1}{3}\): \(1\times\frac{1}{3}=\frac{1}{3}\).

Step2: Find the distance between y - coordinates of R and T

The y - coordinate of R is \(0\), and the y - coordinate of T is \(- 2\). The distance is \(|0-(-2)| = 2\). Multiply by the scale factor \(\frac{1}{3}\): \(2\times\frac{1}{3}=\frac{2}{3}\).

Step3: Calculate the coordinates of \(R'\)

Using the dilation formula \(D_{T,\frac{1}{3}}(x,y)\):
For x - coordinate: \(3+(4 - 3)\times\frac{1}{3}=3+\frac{1}{3}=\frac{9 + 1}{3}=\frac{10}{3}\)
For y - coordinate: \(-2+(0+2)\times\frac{1}{3}=-2+\frac{2}{3}=\frac{-6 + 2}{3}=-\frac{4}{3}\)

Answer:

The coordinates of \(R'\) are \((\frac{10}{3},-\frac{4}{3})\). The distance between the x - coordinates of R and T is \(1\), and the distance between the y - coordinates of R and T is \(2\).