Who is asking: Student
The triangle inequality guarantees that the sum of the lengths of two sides of a triangle is greater than the length of the third. As a consequence, if x and y are legs of a right triangle, with x less than or equal to y, and z the hypotenuse, then x + y is greater than z, so x is greater than z - y. Under what circumstances will x is greater than 2(z - y) be true?
Since you have a right triangle, x2 + y2 = z2 and hence x2 = z2 - y2. Thus if x > 2(z - y) then x2 > 4(z - y)2. Substituting x2 = z2 - y2 gives 0 > 3z2 - 8zy + 5y2. Factor the right side to obtain
0 > (3z - 5y)(z -y)
Since z > y, 3z -5y < 0, that is 0.6z < y. Thus y runs from 0.6z < y < z.
You should now check that the argument in reverse is valid. That is if 0.6z < y < z and x2 + y2 = z2 then x > 2(z - y).
Chris and Penny