https://r2.erweima.ai/mathbot/image/1718357651753ad5exwo9.jpg

Let's analyze the given mathematical problems in the image.

**Problem 3:**
Given a point \( M(a, b) \), and another point \( P(x, y) \), the point \(\Theta(x + a - b, y - a + b)\) represents the relationship with respect to \( P \).

1. For \( P(-6, 7) \) with respect to \( M(2, 3) \), the coordinates of \(\Theta\) are:
   \[
   \Theta = (-6 + 2 - 3, 7 - 2 + 3) = (-7, 8)
   \]

2. Given points \( A(1, -1) \) and \( B(5, -1) \), \( AB \) is a side of the rectangle \( ABCD \). Points \( E(-4, 1) \) and \( F(-2, 2) \), \( G(-1, 0) \) with respect to \( M(4, 4) \) result in points \( E_1, F_1, G_1 \). We need to check if the triangle \( E_1F_1G_1 \) lies within \( ABCD \).

3. For \( P(-1, -1) \), \( N(2, 5) \) with respect to \( M(3, 8) \), coordinates of \( P_1, N_1 \). \( P \) is on the \( x \)-axis and \( N \) on the \( y \)-axis. Determine coordinates of \( N \) by ensuring triangle \( OPN \) area is 3.

**Problem 4:**
Redefine the term "equilibrium point":
If a point \( P \) has coordinates \( (a, b) \), and if the line through \( P \) perpendicular to the \( x \)-axis intersects \( y \)-axis, then the area of the rectangle formed is same as the length of line segments.

1. Determine if point \( P(2, 4) \) is an equilibrium point.
2. For point \( P(a, b) \):
   - \( a = 3, b = 3 \)
   - Determine length of the line segment when extending \( AP \)

Let's solve these problems step-by-step.

Would you like to focus on a specific part or need detailed steps for the entire solution?

Here are 5 questions you might consider:
1. How do we verify the coordinates of transformed points in Problem 3?
2. What is the method to check if a triangle lies within a rectangle?
3. How to find the coordinates of point \( N \) given the area of triangle \( OPN \)?
4. How do we determine if a point is an equilibrium point in Problem 4?
5. What calculations are needed to find the length of line segments in Problem 4?

**Tip:** Always plot geometric figures to better understand the relationships between points and shapes.

Explore solutions to geometric problems involving coordinate transformations and equilibrium points. This detailed analysis covers concepts like coordinate geometry, triangle geometry, and equilibrium points. Suitable for advanced high school students.

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