This is a mathematical problem involving the line $g$ given by the equation $g(x): y = -0.5x - 4$. It consists of four sub-problems, labeled (a) through (d), and deals with rewriting the equation of the line, finding intersections, and deriving parallel and perpendicular lines. Here's a breakdown and explanation of each part:

a) Rewrite the equation in standard and parametric forms:

• Implicit form: Convert $y = -0.5x - 4$ into the form $Ax + By + D = 0$, where $A, B,$ and $D$ are integers.
• Parametric form: Express $x$ and $y$ in terms of a parameter $t$ to describe the line.

b) Intersection with another line and calculation of intersection angle:

• The other line is given as $h: 3x - 6y + 8 = 0$.
• Find the point of intersection of the two lines by solving the system of equations.
• Calculate the angle between the lines using the slopes ($m$) of the lines.

c) Parallel line through point $P(1,1)$:

• Derive the equation of a line parallel to $g$ that passes through the given point. Parallel lines have the same slope as $g$.

d) Perpendicular line through point $P(1,1)$:

• Find the equation of a line perpendicular to $g$ that passes through $P$. Perpendicular lines have slopes that are negative reciprocals of $g$'s slope.

Would you like me to solve each part step-by-step and explain in detail? 😊

Here are 5 related questions to deepen understanding:

1. How do you calculate the angle between two intersecting lines?
2. What are the general forms for parallel and perpendicular lines?
3. How is the parametric form of a line derived from its slope-intercept form?
4. What role does the slope of a line play in determining its relationship to other lines?
5. Why are the coefficients in implicit form $Ax + By + D = 0$ typically integers?

Tip: When finding intersections or transformations of lines, always verify your solution graphically for accuracy!