Example Solution:
y=1/2x, y=-2x, y=1/2x+2, y=-2x+6
Practice makes perfect
To find the equation of four lines having intersections that form a rectangle, we will look for these equations in slope-intercept form.
y=mx+b
Let's see how to guarantee that the intersections form a rectangle.
Opposite sides of a rectangle are parallel, so we need two pairs of parallel lines. Lines are parallel if their slopes are the same.
Consecutive sides of a rectangle are perpendicular, so we need these two pairs of parallel lines to be perpendicular. Lines are perpendicular if their slopes multiplies to - 1.
Let's choose the slopes of the four lines first.
According to the observations above, we need two numbers that multiply to - 1.
We can pick any two such numbers, so let's choose 12 and -2.
Both of these numbers will be the slopes of two of the lines.
y&=1/2x+b_1
y&=1/2x+b_2
y&=- 2x+b_3
y&=- 2x+b_4
This form guarantees that two-two lines are parallel, so the intersections form a parallelogram.
The form also guarantees that the parallel line pairs are perpendicular, so the parallelogram is a rectangle.
Any choice of the y-intercepts will give us four lines we are looking for.
The diagram below shows four such lines and the rectangle they form.
