For this problem, we are given two images, and we need to determine the center of the dilation that happened between them. The first step we need to take, is to determine the ratio of the dilation, for that we will divide the length of one side on image of the dilation by the corresponding side of the original image.
[tex]k=\frac{A^{\prime}B^{\prime}}{AB}=\frac{3}{1}=3[/tex]In order to determine the coordinates of the center, we need to use the following expression:
[tex]\begin{gathered} x_o=\frac{kx_1-x_2}{k-1} \\ y_0=\frac{ky_1-y_2}{k-1} \end{gathered}[/tex]Applying the data from the problem, we have:
[tex]\begin{gathered} x_o=\frac{3\cdot5_{}-5}{3-1}=\frac{10}{2}=5 \\ y_0=\frac{3\cdot2-6}{3-1}=\frac{0}{2}=0 \end{gathered}[/tex]The coordinates of the center are (5,0).