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To determine which lines could represent the other sides of the rectangle given that one side lies along the line shown on the coordinate grid, we need to analyze the properties of rectangles and the equations of lines.

1. **Identify the given line**: The line in the graph passes through the origin (0,0) and has intercepts at (5,5). Therefore, it has a slope \( m = \frac{\Delta y}{\Delta x} = \frac{5}{5} = 1 \). So, the equation of the line is \( y = x \).

2. **Determine the possible orientations of the rectangle's sides**:
    - The opposite side of the rectangle will be parallel to the given line, hence also having a slope of 1.
    - The other two sides of the rectangle will be perpendicular to the given line. The slope of a line perpendicular to a line with slope \( m \) is \( -\frac{1}{m} \). Therefore, for the given line with slope 1, the perpendicular slope will be -1.

3. **Formulate the equations of lines with the identified slopes**:
    - Parallel lines: \( y = x \)
    - Perpendicular lines: \( y = -x \) or any line with a slope of -1

4. **Analyze the provided options**:
    - \( y = -\frac{5}{2}x \): This line has a slope of \(-\frac{5}{2}\), which is not perpendicular or parallel to the given line \( y = x \).
    - \( 5y = 3x - 2 \): This can be rearranged to \( y = \frac{3}{5}x - \frac{2}{5} \), with a slope of \(\frac{3}{5}\), which is neither parallel nor perpendicular to \( y = x \).
    - \( y = \frac{3}{5}x - 5 \): This line has a slope of \(\frac{3}{5}\), which is neither parallel nor perpendicular to \( y = x \).
    - \( 5(y - 2) = 3x \): This simplifies to \( 5y - 10 = 3x \), or \( y = \frac{3}{5}x + 2 \), which has a slope of \(\frac{3}{5}\), which is neither parallel nor perpendicular to \( y = x \).

Since none of the given options represent lines that are parallel or perpendicular to the given line \( y = x \), the answer is that none of the provided lines can form the other sides of the rectangle.

Do you want more details or have any questions about this problem? Here are some related questions for further practice:

1. What is the slope of a line parallel to \( y = -2x + 3 \)?
2. What is the slope of a line perpendicular to \( y = \frac{2}{3}x - 1 \)?
3. How do you determine the equation of a line passing through a given point and parallel to a given line?
4. How do you determine the equation of a line passing through a given point and perpendicular to a given line?
5. How would you graph a rectangle given one of its sides' equations?

**Tip**: When analyzing equations of lines, always start by identifying their slopes as it helps to determine parallel and perpendicular relationships easily.

One side of a rectangle lies along the line shown on the coordinate grid. Along which lines could other sides of the rectangle lie?

Explore how to determine which lines could form the other sides of a rectangle given a line on a coordinate grid. This problem involves key concepts like slope, parallel and perpendicular lines, and their relationships in coordinate geometry. Suitable for Grades 8-10.

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