https://file.aiquickdraw.com/mathbot/file/1727888120298848k7j7w.jpg

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.

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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.

Point P(2, -3) is reflected along different lines. What are the coordinates of the reflected points and the line equations?

This problem explores the reflection of point P(2, -3) across various lines including the x-axis, y-axis, and the origin. Detailed explanations cover reflection formulas, line equations, and intersection points, making it ideal for students in grades 9-10.

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