Write the equations of four lines that intersect to form a square whose sides are not parallel to the axes.
Where would I start, and how would I complete this?
Thank you so much in advance!
To find the equations of the lines that form a square, we need to follow a step-by-step process.
Step 1: Determine the coordinates of the square's vertices.
- Since the sides of the square are not parallel to the axes, let's assume the square has one vertex at the origin (0, 0). We can then choose any side length for the square. For simplicity, let's assume the side length is 2 units.
- Now, we'll label the four vertices as A, B, C, and D, moving counter-clockwise from the origin. Vertex A will be at the origin (0, 0).
Step 2: Determine the equations of the lines passing through each pair of adjacent vertices.
- Line AB: This line connects vertices A and B. The slope of this line can be found as the change in y divided by the change in x. Since A is at (0, 0) and B is at (2, 0), the slope is (0 - 0) / (2 - 0) = 0. The y-intercept of this line is y = 0 since it passes through the x-axis. Therefore, the equation of line AB is y = 0.
- Line BC: This line connects vertices B and C. The slope of this line can be found as the change in y divided by the change in x. Since B is at (2, 0) and C is at (2, 2), the slope is (2 - 0) / (2 - 2) = undefined (division by zero). The x-intercept of this line is x = 2 since it passes through the y-axis. Therefore, the equation of line BC is x = 2.
- Line CD: This line connects vertices C and D. The slope of this line can be found as the change in y divided by the change in x. Since C is at (2, 2) and D is at (0, 2), the slope is (2 - 2) / (0 - 2) = 0. The y-intercept of this line is y = 2 since it passes through the x-axis. Therefore, the equation of line CD is y = 2.
- Line DA: This line connects vertices D and A. The slope of this line can be found as the change in y divided by the change in x. Since D is at (0, 2) and A is at (0, 0), the slope is (0 - 2) / (0 - 0) = undefined (division by zero). The x-intercept of this line is x = 0 since it passes through the y-axis. Therefore, the equation of line DA is x = 0.
Step 3: Verify that the lines intersect correctly.
- We can plot these equations or substitute some sample points to ensure they intersect correctly and form a square.
In summary, the equations of the four lines that intersect to form a square with sides not parallel to the axes are:
- Line AB: y = 0
- Line BC: x = 2
- Line CD: y = 2
- Line DA: x = 0