The side lengths are \( 3x = 15 \), \( 4x = 20 \), and \( 5x = 25 \).

["Solving for Side Lengths in a Classic Right Triangle: Understanding the Equation ( 3x = 15, 4x = 20, 5x = 25 )", "When solving geometric problems involving right triangles, especially those linked to the Pythagorean theorem, equations with side lengths expressed in terms of a variable ( x ) are common and powerful tools. In this article, we explore the linear relationships ( 3x = 15 ), ( 4x = 20 ), and ( 5x = 25 ), revealing how they simplify to find the actual side lengths of a right triangle and reinforcing key concepts in algebra and geometry.", "### What Do the Equations Mean?", "The expressions ( 3x = 15 ), ( 4x = 20 ), and ( 5x = 25 ) represent scaled linear proportions of a triangle’s sides. While side lengths themselves are numerical values (like 15, 20, and 25), these equations allow us to solve for the proportional factor ( x ), which uncovers the real dimensions tied to a right triangle.", "### Solving Each Equation", "Each equation follows the basic algebraic principle of isolating ( x ):", "1. For ( 3x = 15 ):\n Divide both sides by 3:\n [\n x = \frac{15}{3} = 5\n ]", "2. For ( 4x = 20 ):\n Divide both sides by 4:\n [\n x = \frac{20}{4} = 5\n ]", "3. For ( 5x = 25 ):\n Divide both sides by 5:\n [\n x = \frac{25}{5} = 5\n ]", "### The Unified Solution: ( x = 5 )", "All three equations independently yield the same value for ( x ), confirming consistency in the model. This topological and algebraic harmony reflects the defining property of similar triangles and right triangles: all sides scale proportionally from a common factor.", "### From ( x = 5 ) to Real Side Lengths", "Using ( x = 5 ), compute the actual side lengths:\n- ( 3x = 3(5) = 15 )\n- ( 4x = 4(5) = 20 )\n- ( 5x = 5(5) = 25 )", "These numbers ( 15, 20, 25 ) form a well-known Pythagorean triple—a set of integers satisfying ( a^2 + b^2 = c^2 ):\n[\n15^2 + 20^2 = 225 + 400 = 625 = 25^2\n]", "### Why This Matters in Real-World Applications", "Understanding how to solve linear equations derived from geometric principles helps students and professionals alike in fields such as architecture, construction, engineering, and computer graphics. Recognizing proportional relationships allows for quick scaling of models, verifying correctness of constructions, and simplifying problem-solving.", "### Conclusion", "The equations ( 3x = 15 ), ( 4x = 20 ), and ( 5x = 25 ) serve as a concise entry point into applying algebra to geometry. They demonstrate how an unknown multiplier ( x ) brings theoretical side ratios to life, confirming the right triangle with side lengths 15, 20, and 25. Mastery of these foundational problems strengthens analytical skills essential for more advanced mathematics and real-world design challenges.", "---\nKeywords: Pythagoras theorem, right triangle, algebraic equations, proportional reasoning, ( 3x = 15 ), ( 4x = 20 ), ( 5x = 25 ), side length, geometry, algebra, math tutorial, teacher resources, students learning, right triangle ratios", "---", "Summary:\nThe equations ( 3x = 15 ), ( 4x = 20 ), and ( 5x = 25 ) simplify to ( x = 5 ), revealing the proportional side lengths 15, 20, and 25. These values satisfy the Pythagorean theorem, confirming a valid right triangle. Solving such equations builds critical connections between algebra and geometry, forming a foundation for advanced math and practical problem-solving."]