Hey everyone! In this week alone we’ve received two homework help emails about the same problem. We receive lots of emails, but to get two on the same thing within days of each other is rare. Therefore, I assume this particular problem is probably tripping up more than a few of you out there. So I am going to post the problem, and the solution here. Hopefully this will save you a few headaches 
The problem is Geometry Chapter 14 Question 37, dealing with Phone Cells.
Once you have drawn on the triangles posible in this figure (I would suggest you use the diagram for problems 34-37 found in the solutions manual to follow along with this explanation—page 226 at the top.)
He is using the 30-60-90 triangle theorem to solve for line AB (or the distance between two points on this diagram—he chose to call them “cell towers” because real life cell towers are positioned on a grid of hexagons in this way—but that fact, while interesting and great for connecting math to the real world, it is not necessary to understand cell tower grids to answer the problem)
Lets start by identifying where he found a 30-60-90 triangle in this maze of what have been all equilateral triangles up to this point
Look at figure ABC. Notice that inside ABC is 6 triangles – they are all right 30-60-90 triangles. “r” is the hypotenuse of these triangles.
What he does here is, every 30-60-90 triangle has sides that look like this:
Sometimes, the sides will have to be reduced before they look like the ones above, but every single 30-60-90 triangle, will look this way (That means a right triangle where the other two angles of the triangle are 30 degrees and 60 degrees)
So.. Next you have to solve for AC (his answer book used AB—but same process) which corresponds to the hypotenuse of the 30-60-90 triangle.
The solutions guide walks you through to the answer from here, so read over that and write me again if you would like me to walk through it in more detail .
I hope this helps!
