Question 648543: I live 5 miles from the theatre. Tom lives 3 mile from the theatre. How far apart do we live? What is the smallest distance that could separate our homes? What is the largest distance? What are all of the possible distances that could separate our homes?
Answer by swincher4391(1107)   (Show Source):
You can put this solution on YOUR website! 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.
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