SOLUTION: The sides of a triangle are x,y, and sqrt(x^2+xy+y^2). Find its largest angle.

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Question 934926: The sides of a triangle are x,y, and sqrt(x^2+xy+y^2). Find its largest angle.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Since they are lengths of the sides of a triangle,
x%3E0 and y%3E0 .
Then,
sqrt%28x%5E2%2Bxy%2By%5E2%29%3Esqrt%28x%5E2%29=x and
sqrt%28x%5E2%2Bxy%2By%5E2%29%3Esqrt%28y%5E2%29=y .
So, the side measuring z=sqrt%28x%5E2%2Bxy%2By%5E2%29 is the longest side, and the angle opposite that side is the largest angle, which I will call Z .
The law of cosines, applied to a triangle XYZ ,
with side z=XY opposite angle Z , formed by sides x=YZ and y=XZ says
z%5E2=x%5E2%2By%5E2-2%2Ax%2Ay%2Acos%28Z%29
x%5E2%2Bxy%2By%5E2=x%5E2%2By%5E2-2%2Ax%2Ay%2Acos%28Z%29
xy=-2%2Ax%2Ay%2Acos%28Z%29Dividing both sides by xy we get
1=-2cos%28Z%29--->-1%2F2=cos%28Z%29-->Z=120%5Eo or Z=2pi%2F3 .