
Name: Sandra
Who is asking: Student
Level: All
Question:
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?
Hi Sandra,
Since you have a right triangle, x^{2} + y^{2} = z^{2} and hence x^{2} = z^{2}  y^{2}. Thus if x > 2(z  y) then x^{2} > 4(z  y)^{2}. Substituting x^{2} = z^{2}  y^{2} gives 0 > 3z^{2}  8zy + 5y^{2}. 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 x^{2} + y^{2} = z^{2} then x > 2(z  y).
Cheers,
Chris and Penny
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