Let's break down and solve the parts of the problem from the image:

(a) Reflection of Point $P(2, -3)$ to $P'(2, 3)$

Here, point $P(2, -3)$ is reflected to $P'(2, 3)$. The reflection changes the $y$-coordinate while keeping the $x$-coordinate constant. This suggests that the line of reflection is the x-axis. Therefore, $L_1$, the line of reflection, is the x-axis.

Answer for (a): The line of reflection is $L_1: y = 0$ (the x-axis).

(b) Reflection of $P'(2, 3)$ to $P''$

Point $P'$ is reflected along line $L_2$, which is perpendicular to $L_1$ (the x-axis) and passes through the origin. Since $L_2$ is perpendicular to the x-axis, it must be the y-axis. So, we need to reflect $P'(2, 3)$ over the y-axis.

Reflection over the y-axis changes the $x$-coordinate, flipping its sign, while the $y$-coordinate remains unchanged.

Therefore, reflecting $P'(2, 3)$ over the y-axis gives $P''(-2, 3)$.

Answer for (b): The coordinates of $P''$ are $(-2, 3)$.

(c) Point of intersection of $L_1$ and $L_2$

From parts (a) and (b), we know that:

• $L_1$ is the x-axis $y = 0$,
• $L_2$ is the y-axis $x = 0$.

The intersection of the x-axis and y-axis occurs at the origin.

Answer for (c): The point of intersection of $L_1$ and $L_2$ is $(0, 0)$.

(d) Reflection of $P$ to $P'''$

Now, point $P(2, -3)$ is reflected through the origin (the point of intersection from part (c)). Reflecting a point through the origin means both the $x$- and $y$-coordinates are flipped in sign.

Reflecting $P(2, -3)$ through the origin gives $P'''(-2, 3)$.

Answer for (d): The coordinates of $P'''$ are $(-2, 3)$.

Comment on the location of $P''$ and $P'''$

Notice that $P''$ and $P'''$ both have the same coordinates: $(-2, 3)$. This implies that the reflections described in parts (b) and (d) both result in the same point.

Do you want more details or have any questions?

Here are some related questions:

1. What is the geometric interpretation of reflecting over the y-axis?
2. Can you generalize how to reflect a point through any given line?
3. What are the equations of lines perpendicular to the x-axis or y-axis?
4. How can we reflect a point through the origin using vector notation?
5. How would the results change if the line of reflection were not the x- or y-axis?

Tip: Reflections over lines or points can be handled by changing the signs of coordinates, depending on the axis or point of reflection.