Question 648543
Let's consider the case that your house to the theater to Tom is all one line in that order.

Then, it is 5 miles to the theater and then another 3 to Tom's house. So, the maximum distance you could live from Tom is 8 miles.

Let's now consider the case that the line is your house, Tom's house, and then the theater. Then it would be the case that you are (5-3) miles away from Tom or 2 miles. This is the minimum distance.

Now consider that you live 5 miles south of the theater while Tom lives 3 miles east of the theater. You may turn this triangle however you like, I just picked two perpendicular cardinal directions. The case is just a right triangle with legs as known distances.

Then this forms a right triangle. Let a be the distance from your house to the theater: 5 miles. Similarly, let b be the distance from Tom to the theater: 3.

Then a^2 + b^2 = c^2

3^2 + 5^2 = c^2

34 = c^2

c = sqrt(34) = 5.83 miles.

In a similar triangle, consider that the hypotenuse connects you and the theatre, this side is 5. The other leg, Tom to the theater is of distance 3.

Then  a^2 + 3^2 = 5^2

a^2 +3^2 = 5^2 

5^2 - 3^2 = a^2

16 = a^2

a = 4

We can rule out the case that the hypotenuse is the side connecting Tom to the theatre since the hypotenuse is the longest side.

Then to recap...

Then there are 4 possible distances:

2,4,5.83, and 8 miles.

2 is the smallest.

8 is the largest.