To solve this problem, let us analyze it systematically.

Problem Breakdown:

1. The square lies on a coordinate grid.
2. One side of the square lies on the line $y = 4$.
3. Another side lies on the line $y = 9$.
4. A third side lies on the line $x = 12$.
5. We are tasked to find the two possible equations for the fourth side of the square.

Analysis:

• A square has four sides of equal length, and adjacent sides meet at right angles.
• Two opposite sides of the square lie at $y = 4$ and $y = 9$, so the vertical height of the square is $9 - 4 = 5$. Thus, the length of the square is 5 units.
• One side lies on $x = 12$. Since this line is vertical, the fourth side must also be vertical to maintain the square's geometry.

Step-by-Step Solution:

1. The fourth side of the square must be parallel to $x = 12$, so it has the form $x = k$, where $k$ is a constant.
2. The distance between the two vertical sides ($x = 12$ and $x = k$) must equal the side length of the square, which is 5 units.
3. Therefore, $k$ can either be:
   • 5 units to the left of $x = 12$: $k = 12 - 5 = 7$.
   • 5 units to the right of $x = 12$: $k = 12 + 5 = 17$.

Final Answer:

The equations of the two possible lines are: $\boxed{x = 7 \quad \text{and} \quad x = 17.}$

Would you like a detailed diagram or further clarification?

Related Questions:

1. How do we find the length of a square’s side when given only two opposite sides?
2. Can a square exist if three of its sides lie along given lines but the fourth line violates square geometry?
3. What happens if the given lines are not perpendicular—does the figure remain a square?
4. How can we calculate the area of the square from the given problem?
5. Can we generalize this problem to rectangles instead of squares?

Tip:

When working with squares or rectangles on a coordinate grid, always verify that adjacent sides are perpendicular and opposite sides are parallel to maintain the correct geometry.