Question

A smaller commuter airline flies to three cities whose locations form the vertices of a right triangle. The totaly flight distance (from city A to city B to city C and back to city A) is 1400 miles. It is 600 miles between the two cities that are furthest apart. What are the other two distances between the cities?

Answer 1

This requires use of the pythagorean theorem (a^2 + b^2 = c^2). I'll define the distance between cities A and B as a, the distance between cities B and C as b, and the distance between A and C as c. We find that c is the hypotenuse of this triangle if it has the greatest distance (600 miles).

a^2 + b^2 = 600^2
a^2 + b^2 = 360000

We have an issue here. We don't have the distances a and b. To find that, we use the fact that a + b + c = 1400 miles given in the problem. Since c = 600, a + b = 800. Solve for b = 800 - a. Use this as b in the pythagorean theorem.

a^2 + (800 - a)^2 = 360000
a^2 + (800 - a)(800 - a) = 360000
a^2 + 640000 - 1600a + a^2 = 360000
2a^2 - 1600a + 280000 = 0
a^2 - 800a + 140000 = 0
a^2 - 800a = -140000
a^2 - 800a + 160000 = 20000 (completing the square, we take -800/2 = -400 and square that to get 160000; then we add it to both sides)
(a - 400)^2 = 20000
a - 400 = 100sqrt(2)
a = 400 + 100sqrt(2) = 541 miles

b = 800 - (400 + 100sqrt(2)) = 400 - 100sqrt(2)
b = 259 miles

Answer 2

well, whats the distance from city C to city A then? O.o

Answer 3

That is the 600 miles given in the problem.

Answer 4

But on the picture it says the 600 miles is from City A to City B O_O

Answer 5

Oh, I'm sorry. I don't have a picture to look at, so I just chose the 600 miles to be from city C to city A. It can be from city A to city B if that's what the picture says. The 541 miles and 259 miles just represent the other two distances. You'll have to put them where they fit.

Answer 6

Well that makes sense. Thank you so much!