Let the sides be \(a = 13\), \(b = 14\), \(c = 15\). Compute the semi-perimeter \(s\):

["# Let the Sides Be (a = 13), (b = 14), (c = 15): Compute the Semi-Perimeter (s)", "When solving problems in geometry involving triangles, one essential step is calculating the semi-perimeter, denoted as (s). The semi-perimeter is half the sum of a triangle’s three side lengths and serves as a key value in formulas such as Heron’s formula for the area of a triangle. In this article, we focus on a classic 13-14-15 triangle and compute its semi-perimeter (s) step by step.", "## What Is the Semi-Perimeter?", "The semi-perimeter (s) of a triangle is defined as:", "[\ns = \frac{a + b + c}{2}\n]", "This value is critical in Heron’s formula, which calculates the area (A) of any triangle when all three side lengths are known:", "[\nA = \sqrt{s(s - a)(s - b)(s - c)}\n]", "Using (s) simplifies computation and ensures consistency across geometric calculations.", "## Applying the Formula to (a = 13), (b = 14), (c = 15)", "We are given the triangle’s side lengths:", "[\na = 13, \quad b = 14, \quad c = 15\n]", "First, compute the total perimeter:", "[\na + b + c = 13 + 14 + 15 = 42\n]", "Now divide by 2 to find the semi-perimeter:", "[\ns = \frac{42}{2} = 21\n]", "## Significance of (s = 21)", "With (s = 21), we now have a clean value to use in further triangle calculations. For example, Heron’s formula computes:", "[\nA = \sqrt{21(21 - 13)(21 - 14)(21 - 15)} = \sqrt{21 \ imes 8 \ imes 7 \ imes 6}\n]", "This value will yield the triangle’s area:", "[\nA = \sqrt{21 \ imes 8 \ imes 7 \ imes 6} = \sqrt{7056} = 84\n]", "Indeed, the area of the triangle with sides 13, 14, and 15 is 84 square units — thanks to computing (s = 21) first.", "## Conclusion", "Calculating the semi-perimeter is a foundational step in triangle geometry. For the triangle with sides (a = 13), (b = 14), and (c = 15), the semi-perimeter is:", "[\ns = \frac{13 + 14 + 15}{2} = 21\n]", "This value powers advanced computations like Heron’s formula and underscores the importance of precise preliminary steps in geometric problem-solving. Whether for academic purposes or practical applications, knowing how to compute (s) is essential for mastering triangle-related mathematics."]