Knowing two points of a rectangle, how can I figure out the other two?

Question

Hey there guys, I'm learning processing.js, and I've come across a mathematical problem, which I can't seem to solve with my limited geometry and trigonometry knowledge or by help of Wikipedia. I need to draw a rectangle. To draw this rectangle, I need to know the coordinate points of each corner. All I know is x and y for the midpoints of the top and bottom of the box, and the length of all four sides. There is no guarantee on the orientation of the box. Any help? This seems like it should be easy, but it is really stumping me.

Answer 1

The line connecting the midpoints of the sides will be parallel to the top and bottom of the rectangle and the line connecting the midpoints of the top and bottom will be parallel to the two sides of the rectangle.

1. Calculate the slope of the two lines above.
2. Using the point-slope form of a line, write the equation of each side and top and bottom of the rectangle using the slope you calculated and the respective midpoint.
3. Using two of the equation at a time, solve for their intersection point. That will be a corner of the rectangle.

Answer 2

Here are some hints to get you started.

The segment joining the two midpoints must be perpendicular to the top and bottom. If you find the slope of that segment, then the top and bottom must have slopes that are negative/opposite reciprocals. If you write the equations for the lines of the top and bottom of the rectangle then you need to determine what coordinates will be on each line that make your given points the midpoint. By the way you do not really need to be given the width of the rectangle--that value must be the distance between the two given points. Let's say one of the points is given by (a,b) and that the length of the rectangle is given by L. A circle centered at (a,b) with a radius of L/2 must intersect the line representing the top of the box at points that will be two of the corners..

Let's say you calculated the slope of the points joining the two midpoints as m. Then:

y=-1/m(x-a) + b is the equation of the side with midpoint (a,b).

The circle is represented by (x-a)²+(y-b)²= (L/2)²

^{Determine the points of intersection of the line and the circle and you will have the coordinates of two of the corners. Do the same for the other side of the rectangle. You could just as easily use the distance formula rather then the equation of the circle.}

^{Here is a graph that depicts this problem for given midpoints (1,3) and (5,2) with a length of 10.}

https://www.desmos.com/calculator/5ufpg8axwy