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