In Exercises 15 and $16,$ ind the coordinates of any unlabeled vertices. Then nd the indicated length(s).
Find ON and MN.
It can be seen that $N$ is vertically above $D,$ so their $x$ coordinates must be
equal. It is $k$ units above $D,$ therefore the coordinates of $N$ are $(h, k) .$
So if exercise 15 we have the vert sees O and M what was the origin? And is that to H zero and we'll surface first X'd, which is the midpoint between the origin and F Onda Om Andy. We have warden that points for all of which have put on in red. So the first thing we want to do is find the coordinate point for Ed. Now n it has is exactly vertically above D. Therefore, it has the same X value so and must have the same exile uber's d, which is Hey h Andi. It's K units above de Onda as thes on the X axis, but simply que units above the X axis. So the coordinates of n is hate K so we know what final limps O n and N m So we do this by using the lump formula. So if we have two points, the left between them is the square root and then we take the difference of the X coordinates, so x two minus x one square it and then we add this to be squared difference of by coordinates. That's why do mines y one or squared. So let's look. Oh, end. So, Owen, we have the to coordinate points all the origin 00 Onda and which is a h k so square root? The difference in the X coordinates is hate minus zero squared. And we had this to the difference in the wild coordinates, which is K minus zero. Well squared now. I was working with the origin. It's quite easy cause hey, h minus error is simply H and K minus zero is simply K. So the length of o n is h squared plus k squared. Now let's move along to the length of the line and m so And and now the difference in the X coordinates is given by to hate each Marchioness. Hey h because two hatches the X position of M and hate is the experts exposition of n to H minus h or squared. And we had this to the difference of the Y coordinates, which is zero minus k. We'll square now. It doesn't matter that this zero minus K just gives us minus K because when we square it, the minus sign vanishes. We also noticed that to hate minus hate is simply hate. So the length of N m is hate szwed plus k squared. We noticed that this length is exactly the same as the length O n um, this is just interesting because it means that if we were to connect beaver theses Oh, and an M as a triangle, it's a nice sauce Elise Triangle.