Explanation
We are given the following coordinates of a line segment:
[tex]\begin{gathered} C(0,3) \\ D(0,7) \end{gathered}[/tex]We are required to determine the length of the image of the segment CD after being dilated by a factor of 3.
First, we determine the new coordinates of CD after the dilation as:
[tex]\begin{gathered} C(x,y)\to C^{\prime}(3x,3y) \\ C(0,3)\to C^{\prime}(3\times0,3\times3)=(0,9) \\ \\ D(x,y)\to D^{\prime}(3x,3y) \\ D(0,7)\to D^{\prime}(3\times0,3\times7)=(0,21) \end{gathered}[/tex]Therefore, the new coordinates of C and D are C(0, 9) and D(0, 21).
Now, we determine the length of the two points as follows:
[tex]\begin{gathered} C(0,9)\to(x_1,y_1) \\ D(0,21)\to(x_2,y_2) \\ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\ CD=\sqrt{(0-0)^2+(21-9)^2} \\ CD=\sqrt{(0)^2+(12)^2} \\ CD=\sqrt{0+144}=\sqrt{144} \\ CD=12 \end{gathered}[/tex]Hence, the length of the image of the segment CD is 12 units.
