The problem shows a diagonal line representing one side of a rectangle on a coordinate plane. The question asks which lines could represent the other sides of the rectangle. To solve this, we need to analyze the orientation of the given line and find lines that are either parallel or perpendicular to it.

Step-by-Step Breakdown:

1. Determine the slope of the diagonal line:

   • The line passes through points $(0,0)$ and $(5,5)$.
   • Slope $m$ of the line is: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 0}{5 - 0} = 1$ So, the equation of the line is $y = x$ or equivalently, $y - 0 = 1(x - 0)$, which simplifies to $y = x$.

2. Find possible parallel and perpendicular lines:

   • Parallel lines will have the same slope, $m = 1$.
   • Perpendicular lines will have a slope of $-1$, as perpendicular slopes are negative reciprocals of each other.

3. Look at the answer choices:

   • The correct set of equations will include options that describe lines with slopes of either 1 (parallel) or -1 (perpendicular).

Let me know if you'd like me to help match the correct equations from the options or provide further clarification.

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Relative Questions:

1. What is the importance of finding the slope in determining parallel or perpendicular lines?
2. How do you derive the equation of a line given a point and a slope?
3. Why do perpendicular lines have slopes that are negative reciprocals of each other?
4. How does the concept of slope apply to constructing rectangles on a coordinate plane?
5. Can two lines be neither parallel nor perpendicular? What is the relationship in such cases?

Tip:

Always remember that parallel lines have equal slopes, while perpendicular lines' slopes are negative reciprocals. This is crucial for identifying relationships between lines on a graph.