SOLUTION: if x and y are positive numbers with x>y, show that a triangle with sides of lengths 2xy, x^2-y^2, and x^2+y^2 is always a right triangle
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-> SOLUTION: if x and y are positive numbers with x>y, show that a triangle with sides of lengths 2xy, x^2-y^2, and x^2+y^2 is always a right triangle
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Question 430318: if x and y are positive numbers with x>y, show that a triangle with sides of lengths 2xy, x^2-y^2, and x^2+y^2 is always a right triangle Answer by richard1234(7193) (Show Source):
You can put this solution on YOUR website! We know that must be the longest side of the triangle (it cannot be , I will demonstrate why afterwards). By the Pythagorean theorem,
. Therefore this is a right triangle.
The only "assumption" we had to make was that , but this is easy to prove. By the AM-GM inequality, . Multiplying both sides by 2, . Equality occurs only when , but this cannot be true, so is strictly greater than .
