Let’s say you are lost and trying to locate yourself with your phone. What is happening is that your phone sends out a signal and records how long it takes to make contact with a receiver and then uses this information to calculate the distance between you and the receiver. The goal is to make contact with three receivers, let’s call them A, B, and C:

For us, it is important to have a frame of reference, so let’s add axes and some values assuming we know the locations of the receivers:

Obviously, you can’t see this, but this is what your GPS is doing. From here, distances from A, B, and C are known, but this is not enough because for example, if the distance from A is 9.489 miles, then it could be on any point of the circle around A with radius 9.489:

But now let’s include B, which is found to be 9.487 miles away:

Notice how this has narrowed our location down to two possible points (where the two circles overlap. Now, C is the tiebreaker, which is found to be 14.422 miles away:

Notice how the three circles only intersect at one point (me), therefore the GPS has found our location in relation to A, B, and C. This is the conceptual idea, but there is some number crunching to make this happen. The equations of the circles are of the following form:

Where (h, k) is the center of the circle and r is the radius. This gives us the following equations for each circle:

Starting with A and B, I am going to rearrange their equations:

Now I can set them equal to each other and solve for x:

Though I could use either equation, I am going to substitute into the equation for circle A to find the y-value:

So this means the points of intersection are (4, 9) or (4, -9). Rather than doing a ton of work again, I am simply going to substitute both points into the equation for circle C to see which one works:

And there we have it, our location must be at (4,9), which is correct:

Check out the following possible research tasks:

- There Has Got to Be An Easier Way: Students will be pushed to prove a trilateration formula for two-dimensions, then see if it holds for three-dimensions.
- Oh No! What About Elevation!?!: Based on the last research task, students will investigate the formulas for three-dimensions.