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.