Problem:


Consider a triangle where the lengths of the sides are \(a\), \(b\), and \(c\). Suppose the semi-perimeter of the triangle is \(s\), defined as:


\[ s = \frac{a + b + c}{2} \]


The area \(A\) of the triangle can be calculated using Heron's formula:


\[ A = \sqrt{s(s - a)(s - b)(s - c)} \]


Given that the semi-perimeter \(s\) is 15 units, and the area \(A\) is 60 square units, find the possible integer values for the sides \(a\), \(b\), and \(c\) of the triangle.


Solution Outline:


1. Start with the formula for the semi-perimeter: \( s = \frac{a + b + c}{2} \). Given \( s = 15 \), it follows that \( a + b + c = 30 \).


2. Substitute \( s \) and \( A \) into Heron's formula to get:


   \[ 60 = \sqrt{15(15 - a)(15 - b)(15 - c)} \]


3. Square both sides to eliminate the square root:


   \[ 3600 = 15(15 - a)(15 - b)(15 - c) \]


4. Simplify the equation:


   \[ 240 = (15 - a)(15 - b)(15 - c) \]


5. Determine integer solutions for \(a\), \(b\), and \(c\) that satisfy both the sum \(a + b + c = 30\) and the equation \(240 = (15 - a)(15 - b)(15 - c)\).


6. Solve for possible integer values.


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