Question

1. the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

a. for the triangle shown, anna said that any value of x greater than 5 is possible. explain anna’s error.
triangle with sides 6, 11, and x (11 as base, 6 and x as the other two sides)

Explanation:

Step1: Apply triangle inequality theorem

The triangle inequality theorem states that for a triangle with sides \(a\), \(b\), and \(c\), the following must hold: \(|a - b| < c < a + b\). Here, the two given sides are \(6\) and \(11\), and the third side is \(x\). So we have two inequalities:

1. \(x>11 - 6\) (because the length of a side must be greater than the difference of the other two sides)
2. \(x<11 + 6\) (because the length of a side must be less than the sum of the other two sides)

Step2: Solve the inequalities

First, solve \(x>11 - 6\):
\(11-6 = 5\), so \(x > 5\).
Second, solve \(x<11 + 6\):
\(11 + 6=17\), so \(x < 17\).

Combining these two inequalities, we get \(5 < x < 17\). Anna only considered the lower bound (\(x>5\)) but did not consider the upper bound (\(x < 17\)). So values of \(x\) must be greater than \(5\) AND less than \(17\), not just any value greater than \(5\) (for example, \(x = 20\) would not form a valid triangle with sides \(6\) and \(11\) because \(6+11=17<20\)).

Answer:

Anna's error is that she only considered the lower bound (\(x > 5\)) from the triangle inequality theorem but ignored the upper bound (\(x<17\)). The correct range for \(x\) is \(5 < x < 17\), so \(x\) must be greater than \(5\) AND less than \(17\), not just any value greater than \(5\).