Let's analyze and solve the problem step by step.

Part (a):

The problem states that point $P(2, -3)$ is reflected to become $P'(2, 3)$. We are asked to name the line of reflection, say $L_1$.

Solution:

When reflecting a point, the line of reflection lies halfway between the original point and its image, and both are equidistant from this line. In this case, the points $P(2, -3)$ and $P'(2, 3)$ differ only in their $y$-coordinates, which indicates a reflection over a horizontal line. Specifically, the line of reflection is the x-axis (since the x-axis lies at $y = 0$).

Thus, $L_1$ is the x-axis.

Part (b):

Point $P'$ is reflected to $P''$ along a line $L_2$, which is perpendicular to the line $L_1$ (the x-axis) and passes through the point invariant along both axes.

Solution:

A line perpendicular to the x-axis is a vertical line. The point invariant along both axes is the origin (0,0), because the origin does not change under reflections over either axis. Therefore, $L_2$ is the line x = 0 (the y-axis).

The reflection of point $P'(2, 3)$ across the y-axis involves changing the sign of its $x$-coordinate while keeping the $y$-coordinate the same. So, $P''$ is:

$P'' = (-2, 3)$

Part (c):

We are asked to name and write the coordinates of the point of intersection of the lines $L_1$ and $L_2$.

Solution:

The lines $L_1$ (the x-axis, $y = 0$) and $L_2$ (the y-axis, $x = 0$) intersect at the origin (0, 0).

Part (d):

Point $P$ is reflected to $P'''$ on reflection through the point named in part (c), which is the origin. We need to write the coordinates of $P'''$ and comment on the location of the points $P'$ and $P'''$.

Solution:

Reflecting a point through the origin involves negating both the $x$- and $y$-coordinates. Therefore, reflecting $P(2, -3)$ through the origin gives:

$P''' = (-2, 3)$

Comment: The point $P'''$ is the same as $P''$, which was found in part (b). This suggests that reflecting $P$ through both the x-axis and y-axis sequentially results in the same point as reflecting $P$ directly through the origin.

Summary:

• (a) The line of reflection $L_1$ is the x-axis.
• (b) The coordinates of $P''$ are $(-2, 3)$.
• (c) The intersection point of $L_1$ and $L_2$ is the origin $(0, 0)$.
• (d) The coordinates of $P'''$ are $(-2, 3)$.

Would you like further details or have any questions on these steps?

Here are 5 related questions for further exploration:

1. How do you find the line of reflection between two points?
2. What properties of a point are preserved under reflection across the x-axis?
3. How can you geometrically construct the reflection of a point over a line?
4. What transformations result in symmetry with respect to the origin?
5. How do you determine if two reflections yield the same result as a direct transformation?

Tip: Reflections over the x-axis change the sign of the y-coordinate, while reflections over the y-axis change the sign of the x-coordinate.