Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school
Brian was a geometry teacher through the Teach for America program and started the geometry program at his school
Given a rectangle drawn in a coordinate plane, it is possible to find the point of intersection of the diagonals.
First, write the vertices in terms of the given side lengths. If the bottom left vertex is (0,0), and the left side has length a, then the upper left vertex is at the coordinate (0,a). Then, given a horizontal length of b + c, the upper right vertex is at (b + c,a), and moving down length a, the lower right vertex is (b + c,0).
Then, recall that the diagonals of a rectangle bisect one another. So, find the midpoint of one of the diagonals formed in this rectangle (the endpoints of the diagonal are the coordinates of opposite sides).
Sometimes in geometry we like to take a coordinate system and apply it to what we know. In this problem it says the quadrilateral is a rectangle. Find the point of intersection of the diagonals.
So the first thing I’m going to do is I’m going to draw in my diagonals just to give us an idea of what we’re looking at. And what do I know about the diagonals of a rectangle? Well, if I look at this diagram here we see that the diagonals are congruent and that they bisect each other. So this key part about bisecting each other is going to help us find this point of intersection.
So let’s start off by writing the ordered pairs at each of the four vertices. Well, we know that this is at (0,0) and if this has a length of A, we know that this is going to be a vertical distance, so that’s going to be at point (0,a) because we haven’t moved left or right on our X axis. If I look at this point right here, we’ve travelled a distance of b plus c. How do I know that? Because in a rectangle opposite sides are going to be congruent because it’s a parallelogram.
So we’re going to move b plus c to the right, so I’m going to write b plus c but we’ve not moved up or down so our y coordinate is going to stay the same. So that’s going to be zero.
Now our final coordinate is going to be we’ve moved over b plus c and we’ve gone up a. So I know that if these are bisected, then I’m going to have to find the midpoint of my sides. The midpoint is just finding the total distance and dividing by two. So we know that the total distance that we’re moving, x is b plus c.
So if I say the intersection, not midpoint but the intersection of the diagonals is going to be however far we’ve moved down the x divided two, we moved b plus c divided by 2, and we’ve gone up a but since it’s bisected, our intersection will be at a divided by 2. Make that a little shorter to it looks like an a and not a Q.
So the key thing here is remembering that the diagonals bisect each other and labelling our vertices using what we know about the coordinate system.