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

The formula for the distance between two x - coordinates \(x_1\) and \(x_2\) is \(|x_1 - x_2|\).
For point \(R(0,4)\) and \(T(3,-2)\), \(|0 - 3|=3\)

Step2: Find the distance between y - coordinates

The formula for the distance between two y - coordinates \(y_1\) and \(y_2\) is \(|y_1 - y_2|\).
For point \(R(0,4)\) and \(T(3,-2)\), \(|4-(-2)| = |4 + 2|=6\)

Step3: Find the distance of \(R'\) from \(T\)

Since the scale factor \(k=\frac{1}{3}\), the distance of \(R'\) from \(T\) is \(k\times\) (distance from \(R\) to \(T\)).
Using the distance formula \(d=\sqrt{(x_1 - x_2)^2+(y_1 - y_2)^2}\) for \(R(0,4)\) and \(T(3,-2)\), \(d=\sqrt{(0 - 3)^2+(4 + 2)^2}=\sqrt{9 + 36}=\sqrt{45}=3\sqrt{5}\). After dilation with scale factor \(k = \frac{1}{3}\), the distance of \(R'\) from \(T\) is \(\frac{1}{3}\times3\sqrt{5}=\sqrt{5}\)
Another way: Using the property of dilation. If we consider the vector from \(T\) to \(R\) as \(\overrightarrow{TR}=(0 - 3,4+ 2)=(-3,6)\). After dilation with scale factor \(k=\frac{1}{3}\), the vector \(\overrightarrow{TR'}=\frac{1}{3}\overrightarrow{TR}=(-1,2)\)

Step4: Find the coordinates of \(R'\)

If the center of dilation is \(T(3,-2)\) and \(\overrightarrow{TR'}=(x_{R'}-3,y_{R'}+2)\)
Since \(\overrightarrow{TR'}=(-1,2)\), we have \(x_{R'}-3=-1\) and \(y_{R'}+2 = 2\)
Solving \(x_{R'}-3=-1\) gives \(x_{R'}=2\), solving \(y_{R'}+2 = 2\) gives \(y_{R'}=0\)

Answer:

The distance between the \(x\) - coordinates of \(R\) and \(T\) is \(3\).
The distance between the \(y\) - coordinates of \(R\) and \(T\) is \(6\).
\(R'\) is \(\sqrt{5}\) (or \(1/3\) of the distance from \(R\) to \(T\)) from \(T\), so the coordinates of \(R'\) are \((2,0)\)