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Coordinates and Distance Formula, a la Shmoop. Frodo has acquired the One Donut of Power,
and now he must travel across Middle Earth to Mount Doom to destroy it.
It's a long and harrowing journey, so good thing he has a few buddies to keep him company
along the way.
Uh, maybe. They all start out in Hobbiton, which is 13
leagues west and 19 leagues north of Rohan, Middle Earth's center.
Mount Doom is located exactly 8 leagues east and 9 leagues south of Rohan.
How far do Frodo and his friends have to travel on their quest to destroy the donut?
Rather than use a pedometer to find the distance…
…we can make use of coordinates and the distance formula.
Let's look at Middle Earth as a coordinate plane.
Using Rohan as the origin and north, south, east, and west as our axes…
…we can find the coordinates of Hobbiton and Mount Doom.
Hobbiton is 13 leagues west and 19 leagues north of Rohan…
...so its coordinates are negative 13, 19. Mount Doom is 8 leagues east and 9 leagues
south of Rohan...
…so its coordinates are 8, negative 9. Now that we have the coordinates of our two
…we can calculate the distance between them using the one formula to rule them all:
the distance d equals the square root of x2 minus x1 squared plus y2 minus y1 squared…
…where x1, y1 and x2, y2 are the coordinates of the two endpoints.
Frodo and his friends start out in Hobbiton, so that's our first point.
x1 equals negative 13 and y1 equals 19.
They want to get to the other end point, Mount Doom.
So x2 equals 8 and y2 equals negative 9. If we plug these numbers into our formula…
…we should get d equals the square root of 8 minus negative 13 squared plus negative
9 minus 19 squared.
Simplifying that all the way will get us d equals 35.
So the distance from Hobbiton to Mount Doom is exactly 35 leagues. How long that'll take…
Well, that's a different matter.