SOLUTION: A smaller commuter airline flies to three cities whose locations form a vertices of a right triangle. The total flight distance( from city A to city B to city C and back to city A)

Algebra ->  Pythagorean-theorem -> SOLUTION: A smaller commuter airline flies to three cities whose locations form a vertices of a right triangle. The total flight distance( from city A to city B to city C and back to city A)      Log On


   

Question 586647: A smaller commuter airline flies to three cities whose locations form a vertices of a right triangle. The total 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. Find the other two distances between cities.....
Found 2 solutions by zoomkaboom4, Alan3354:
Answer by zoomkaboom4(62) About Me  (Show Source):
You can put this solution on YOUR website!
Since it is a right triangle, the other two cities are 400 miles apart because..
400+400+600 = 1400 miles.

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
A smaller commuter airline flies to three cities whose locations form a vertices of a right triangle. The total 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. Find the other two distances between cities
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The hypotenuse = 600 miles
The other 2 legs, a & b, add to 800 miles
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a + b = 800
b = 800 - a
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a%5E2+%2B+b%5E2+=+600%5E2
a%5E2+%2B+%28800-a%29%5E2+=+600%5E2
2a%5E2+-+1600a+%2B+640000+=+360000
a%5E2+-+800a+%2B+140000+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-800x%2B140000+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-800%29%5E2-4%2A1%2A140000=80000.

Discriminant d=80000 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--800%2B-sqrt%28+80000+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-800%29%2Bsqrt%28+80000+%29%29%2F2%5C1+=+541.42135623731
x%5B2%5D+=+%28-%28-800%29-sqrt%28+80000+%29%29%2F2%5C1+=+258.57864376269

Quadratic expression 1x%5E2%2B-800x%2B140000 can be factored:
1x%5E2%2B-800x%2B140000+=+%28x-541.42135623731%29%2A%28x-258.57864376269%29
Again, the answer is: 541.42135623731, 258.57864376269. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-800%2Ax%2B140000+%29