Question

△rst is dilated with the rule dt,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 between them is \(|4 - 3|=1\).

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 between them is \(|0-(-2)| = 2\).

Step3: Use the dilation rule \(D_{T,\frac{1}{3}}(x,y)\)

The formula for dilation with center \((a,b)\) and scale factor \(k\) is \((x',y')=(a + k(x - a),b + k(y - b))\). Here \(a = 3\), \(b=-2\), \(k=\frac{1}{3}\), \(x = 4\), \(y = 0\).
For the x - coordinate of \(R'\): \(x'=3+\frac{1}{3}(4 - 3)=3+\frac{1}{3}=\frac{9 + 1}{3}=\frac{10}{3}\)
For the y - coordinate of \(R'\): \(y'=-2+\frac{1}{3}(0+2)=-2+\frac{2}{3}=\frac{-6 + 2}{3}=-\frac{4}{3}\)

Answer:

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