Problem 60 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. In Heron's Area Formula, \(s\) is the average of the lengths of the three sides of the triangle.

Step-by-Step Solution

Answer

The statement is false. In Heron's Area Formula, \(s\) represents the semiperimeter, not the average, of the lengths of the three sides of a triangle.

Step 1: Understand Heron's Formula

Heron's Area Formula is used to calculate the area of a triangle with sides of lengths \(a\), \(b\), and \(c\). The formula is \(A = \sqrt{s(s - a)(s - b)(s - c)}\) where \(s\) is known as the semiperimeter of the triangle.

Step 2: Define the Semiperimeter of the Triangle

By definition, in the context of Heron's formula, \(s\) is the semiperimeter of the triangle and is calculated as \(s = \frac{a + b + c}{2}\).

Step 3: Evaluate the Statement

The statement says that \(s\) is the average of the lengths of the three sides of the triangle. A direct average would divide the sum of \(a\), \(b\), and \(c\) by 3, not 2, so the statement is not correct.