Question

a student said that a triangle with side lengths of 3 ft and 4 ft could have a third side with a length of 7 ft. explain why the student is incorrect. what is a correct statement about the third side of the triangle?
by the triangle inequality theorem, the length of the third side must be
(square) ft because the sum of two side lengths must be
(square) the length of the third side. the third side must be less than (square) ft but greater than (square) ft.

Explanation:

Step1: Apply the triangle inequality theorem

The triangle inequality theorem states that for a triangle with side lengths \(a\), \(b\), and \(c\), the following must hold: \(a + b>c\), \(a + c>b\), and \(b + c>a\). Let \(a = 3\) and \(b=4\). Then \(a + b>c\) gives \(3 + 4>c\) (i.e., \(c<7\)), and \(|a - b|

Step2: Analyze the student's claim

The student claims that \(c = 7\). But from \(a + b>c\) (\(3+4 = 7\)), when \(c = 7\), \(a + b=c\), which does not satisfy the strict inequality \(a + b>c\) required for a non - degenerate triangle.

Answer:

The length of the third side must be less than \(7\) ft but greater than \(1\) ft. The student is incorrect because when \(c = 7\), \(3+4=7\) and the triangle inequality \(a + b>c\) (where \(a = 3\), \(b = 4\), \(c\) is the third side) is not satisfied. A correct statement is: The length of the third side of the triangle must be less than \(7\) ft but greater than \(1\) ft.