 In this video, we're going to write a coordinate proof, proving that if a quadrilateral is a parallelogram, then its diagonals bisect each other. So in order to do this, we're going to first write down the given. And the given is that the quadrilateral is a parallelogram. Now the parallelogram that we're using is Owen. So we are going to say parallelogram Owen is the given. What we're trying to prove tells us right here, the diagonals bisect each other. In order to write down what we're trying to prove, let's stop for a second and actually put Owen onto the graph. The coordinates are on your note sheet, so you might pause the video for a second and put Owen on your graph, and I will do the same. Okay, so your picture should look like this with the points O, W, E, N. And what we're trying to prove is that the diagonals bisect each other. So what I'm going to do next is I'm going to draw the diagonals from O to E and from W to N. And what you'll notice, if you do maybe a little better job than I just did, is that those two segments intersect at this point. And this point, we're going to just give it a name. It doesn't matter what letter you use, so let's just go ahead and call it X. And we can count and see that that point X is the ordered pair 3, 2. So what we're going to do is we are going to prove that the diagonals bisect each other, which means that WX has to be congruent to NX and EX has to be congruent to OX. And so that's what we're going to write here in the prove, is that we want to show that WX is congruent to NX. And we also want to show that EX is congruent to OX because we know the word bisect means cut in half or cut into two equal pieces. So we know when we draw the diagonals, if they do in fact bisect each other, it means that these two pieces have to be congruent and these two pieces have to be congruent. Now in order to actually do the proof, you're going to find the length of each of these four segments. And hopefully what you'll find is that they are in fact equal. In order to find the length, we are going to use the distance formula. So we're going to, over where it says proof, we are going to start with finding the distance from W to X. And so using the distance formula and using the points 0, 4, and 3, 2, because those are the points for W and X, we are going to get 2 minus 4 squared plus 3 minus 0 squared. And if we evaluate that, 2 minus 4 is negative 2. When you square it, you get 4. And then 3 minus 0 is 3. When you square it, you get 9. Add those together and it's the square root of 13. So now what we want to do is find NX and our hope is that it is going to equal the square root of 13. So once we do that, we can see that sure enough, using the points N and X, we get 2 minus 0 squared plus 3 minus 6 squared, which does come out to equal the square root of 13. So we have proven that WX is congruent to NX. Next we're going to do the same thing, but we are going to do the distance formula with the EX and the OX. Using the points 6, 4, and 3, 2 distance formula, we find out that EX is equal to square root of 13. And so lastly we want to find OX and we're going to use the points 0, 0, and 3, 2 to find OX. And once again, using those two points, we figure out that OX is equal to the square root of 13. So we have now proven the other piece, which is EX is congruent to OX. And so the final thing you're going to do is write a conclusion. And what we can conclude is if we look back up here, we can conclude that OIN, since OIN is a parallelogram, the diagonals do in fact bisect each other.