SOLUTION: 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 di

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.

RELATED QUESTIONS
Jones lives 5 miles from Brown. Brown lives 12 miles from Smith. What is the shortest... (answered by checkley79)

DENISE,KEITH AND TIM LIVE IN THE SAME NEIGHBORHOOD.DENISE LIVES 0.3 MILES FROM KEITH.THE... (answered by solver91311)

Lisa lives 25 miles due south of the mall. Lisa also lives 27 miles due east of her... (answered by ikleyn)

Teddy and Britt live 8mi apart from each other, but in opposite directions from the... (answered by josmiceli)

frank lives 3/10 mile from school. michelle lives 2 1/3 times as far from school as frank (answered by rfer,checkley79,solver91311)

A school provides bus service only to students who live a distance greater than 2 miles... (answered by greenestamps,Boreal)

A school provides bus service only to students who live a distance greater than 2 miles... (answered by ewatrrr)

I'm confused as on how to solve this problem. I tried doing a^2 + b^2 = c^2, but when I... (answered by Alan3354,josgarithmetic)

Joseph live 5 miles from the school and Maya lives 2 miles from the school. If Joseph... (answered by ptaylor)