Question

dilating a figure with the center of dilation at a point
△rst is dilated with the rule ( d_{t,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 \(d_x=\vert4 - 3\vert=1\).

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 \(d_y=\vert0-(-2)\vert = 2\).

Step3: Apply the dilation rule

The dilation rule is \(D_{T,\frac{1}{3}}(x,y)\).
For the x - coordinate of \(R'\):
Let \(x_R = 4\), \(x_T=3\). Using the formula \(x'=x_T+\frac{1}{3}(x_R - x_T)\)
\(x'=3+\frac{1}{3}(4 - 3)=3+\frac{1}{3}=\frac{9 + 1}{3}=\frac{10}{3}\)
For the y - coordinate of \(R'\):
Let \(y_R = 0\), \(y_T=-2\). Using the formula \(y'=y_T+\frac{1}{3}(y_R - y_T)\)
\(y'=-2+\frac{1}{3}(0-(-2))=-2+\frac{2}{3}=\frac{-6 + 2}{3}=-\frac{4}{3}\)

Answer:

The coordinates of \(R'\) are \((\frac{10}{3},-\frac{4}{3})\)