This problem is a combination of two mathematical principles: the distance formula between two points, and the reduction of radicals.

To begin this problem we must find the lengths of all sides of the triangle. Because we have the coordinates of the end points of each side, we can apply the distance formula.

  = distance between the two points  and .

The application of this formula by writing out all the symbols and inserting the points and perhaps using a calculator is cumbersome and can be time consuming. There is a faster application of this formula in a geometric sense. Follow the steps below to give this process a try. This should be repeated for each distance you are trying to find and in our case, it is three different distances.

1)  Draw a right triangle with one leg horizontal and one vertical.

2)  For the horizontal leg, find the distance between the  coordinates of the two choosen points and write down that distance: .

3)  Repeat for the vertical leg substituting the  coordinates: .

4)  Apply the Pythagorean Theorem, where  and  are the horizontal and vertical leg in no particular order.

5)   Solve for .

 is the length of one of the sides of your triangle in which you wish to find the perimeter. If the formula is faster for you, use it. If the geometric method is faster and easier to visualize, use that one instead.

After applying the formula to all three legs, you’ll find that you have the lengths, , , and . Once you add them together to find the perimeter (the perimeter of a triangle is all sides added together), you have the value . Unfortunately you won’t find the answer as you still need to simplify the radical.

To go about reducing the radical you will need to break down 52 into its prime factors: 13, 2, and 2. (As an aside, any even number can be checked for prime factors by dividing it by two until it is no longer even.) Since there are two 2’s under the radical, we can rewrite the radical as  or . Since  we can reduce  to .

Therefore,  can be rewritten  or , which is our perimeter. 

What is the distance of the line

Between  and ?

To calculate this, you need to know the points for these two values.  To find these, substitute in for the two values of  given:

Likewise, do the same for :

Now, this means that you have two points:

 and 

The distance formulat between two points is:

For our data, this is:

This is:

 or approximately 

In order to find the distance between these two points, we first need the points!  To find them, substitute in  for x and y.  Thus, for each you get:

Thus, the two points are:

 and 

The distance formulat between two points is:

For your data, this is very simply:

 or 

To find the distance between two points, use the distance formula:

this gives .

In order to find the actual distance between these two towns, we must find the distance between these two points. We can find the distance between the points by using the distance formula, where the variable, , represents the distance:

 The variables  and  represent the x-values at each point and  and  represent the y-values at each point. We can calculate the distance by substituting the x- and y-coordinates of each point into the distance formula. 

Since each unit on the map represents an actual distance of  miles, the actual distance between the two cities can be calculated using the the following operation: