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### Question 88:

#### (a) **Reflection of Point P (2, -3) to P' (2, 3):**
The point P (2, -3) is reflected to P' (2, 3). The coordinates suggest that the reflection has occurred over the **x-axis**, because the x-coordinate remains unchanged (2), and the y-coordinate has changed sign from -3 to 3.

**Answer for (a):** The line of reflection \( L_1 \) is the **x-axis**.

#### (b) **Reflection of P' (2, 3) to P'' along a perpendicular line (L_2):**
The line \( L_2 \) is described as perpendicular to \( L_1 \) (the x-axis) and passes through a point that is invariant along both axes. This means the line passes through the **origin** (0,0), because the origin is the only point that is invariant along both the x-axis and y-axis.

Since \( L_2 \) is perpendicular to the x-axis, it must be the **y-axis**. Reflecting P' (2, 3) over the y-axis means the x-coordinate will change sign while the y-coordinate remains the same. Thus, the reflection of P' (2, 3) across the y-axis results in \( P'' (-2, 3) \).

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

#### (c) **Point of intersection of \( L_1 \) (x-axis) and \( L_2 \) (y-axis):**
The x-axis (line \( L_1 \)) and the y-axis (line \( L_2 \)) intersect at the origin.

**Answer for (c):** The point of intersection of \( L_1 \) and \( L_2 \) is the **origin (0, 0)**.

#### (d) **Reflection of P to P''' through the origin:**
Point P (2, -3) is reflected through the origin. Reflecting a point through the origin involves changing the sign of both the x- and y-coordinates. Therefore, reflecting P (2, -3) through the origin gives \( P''' (-2, 3) \).

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

**Comment on P'' and P'''**: Both \( P'' \) and \( P''' \) are at the same location, i.e., (-2, 3).

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Here are 5 related questions for deeper understanding:
1. What is the general formula for reflecting a point over the x-axis?
2. How do you find the equation of a line perpendicular to another line?
3. What properties of the origin make it invariant under both x- and y-axis reflections?
4. How does reflecting a point over the origin differ from reflecting over an axis?
5. Can you describe how to reflect a point over any arbitrary line?

**Tip:** When reflecting over the x-axis or y-axis, only the respective coordinate changes sign, while reflecting through the origin changes both coordinates.

Point P (2, -3) on reflection becomes P'(2,3). Name the line of reflection (say L1). (b) Point P' is reflected to P'' along the line (L2), which is perpendicular to the line L1 and passes through the point, which is invariant along both axes. Write the coordinates of P''. (c) Name and write the coordinates of the point of intersection of the lines L1 and L2. (d) Point P is reflected to P''' on reflection through the point named in the answer of part I of this question. Write the coordinates of P'''. Comment on the location of the points P'' and P'''.

Explore how to reflect point P (2, -3) across the x-axis, y-axis, and origin through multiple reflections in coordinate geometry. Learn step-by-step solutions, key concepts of symmetry, and properties of reflection. Suitable for students in Grades 9-10.

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