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?

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

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).