Question

△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 the \(x -\)coordinates of \(R\) and \(T\)

The \(x -\)coordinate of \(R\) is \(0\), and the \(x -\)coordinate of \(T\) is \(3\). The distance is \(|3 - 0|=3\).

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

The \(y -\)coordinate of \(R\) is \(4\), and the \(y -\)coordinate of \(T\) is \(- 2\). The distance is \(|4-(-2)| = |4 + 2|=6\).

Step3: Use the dilation formula

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

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 \(\frac{1}{3}\) the distance of \(RT\) from \(T\), so the coordinates of \(R'\) are \((2,0)\).