Certainly! Let's go through the steps to determine the coordinates of another vertex of the figure, given that the figure has line symmetry about the [tex]\( y \)[/tex]-axis.
1. Understanding Symmetry about the [tex]\( y \)[/tex]-axis:
Line symmetry about the [tex]\( y \)[/tex]-axis means that for any point [tex]\((x, y)\)[/tex] on the figure, there should be a corresponding point [tex]\((-x, y)\)[/tex] on the opposite side of the [tex]\( y \)[/tex]-axis. This symmetry reflects the x-coordinate across the [tex]\( y \)[/tex]-axis while keeping the y-coordinate unchanged.
2. Given Vertex:
We are given a vertex of the figure at [tex]\((-1, -3)\)[/tex].
3. Determine the Symmetric Vertex:
To find the vertex that is symmetric to [tex]\((-1, -3)\)[/tex] about the [tex]\( y \)[/tex]-axis, we need to change the sign of the x-coordinate while keeping the y-coordinate the same:
[tex]\[
\text{symmetric vertex} = (-(-1), -3) = (1, -3)
\][/tex]
4. Verify the Correct Answer:
We now check the options provided and find that the coordinate [tex]\( (1, -3) \)[/tex] matches one of the choices.
Therefore, taking all the steps and reasoning into account, the coordinates of another vertex of the figure that satisfies the condition of line symmetry about the [tex]\( y \)[/tex]-axis are [tex]\((1, -3)\)[/tex].
[tex]\[
\boxed{(1, -3)}
\][/tex]
