By the Pythagorean theorem: \( 5^2 + b^2 = 13^2 \).

["Solving Pythagorean Theorem Problems: Using ( 5^2 + b^2 = 13^2 )", "Understanding the Pythagorean theorem is essential for solving right triangle problems. One classic equation derived from this theorem is:", "[\n5^2 + b^2 = 13^2\n]", "This equation represents a right triangle where 5 and ( b ) are the legs, and 13 is the hypotenuse. Let’s break down how to solve for ( b ) step by step and explore the significance of this formula in geometry and real-world applications.", "---", "### What Is the Pythagorean Theorem?", "The Pythagorean theorem states that in a right triangle:", "[\na^2 + b^2 = c^2\n]", "where ( c ) is the length of the hypotenuse (the side opposite the right angle), and ( a ) and ( b ) are the lengths of the other two sides. Applying this to our equation:", "- ( a = 5 )\n- ( c = 13 )\n- ( b ) is the unknown leg", "---", "### Step-by-Step: How to Solve for ( b )", "1. Substitute known values into the formula:", "[\n5^2 + b^2 = 13^2\n]", "2. Calculate the squares:", "[\n25 + b^2 = 169\n]", "3. Isolate ( b^2 ) by subtracting 25 from both sides:", "[\nb^2 = 169 - 25 = 144\n]", "4. Take the square root of both sides:", "[\nb = \sqrt{144} = 12\n]", "---", "### Final Answer", "[\nb = 12\n]", "This means the missing leg of the triangle measures 12 units in length.", "---", "### Why This Equation Matters", "Using ( 5^2 + b^2 = 13^2 ) illustrates the core principle of the Pythagorean theorem: verifying whether a triangle is right-angled or solving for an unknown side. This theorem is foundational in:", "- Geometry: Proving triangles, calculating distances\n- Trigonometry: Relating side ratios to angles\n- Architecture & Engineering: Ensuring structural accuracy\n- Physics: Determining resultant vectors", "---", "### Quick Tips When Applying the Theorem", "- Always assign the longest side as the hypotenuse ( c ).\n- Verify your calculations carefully—small errors in squaring or taking square roots can lead to wrong results.\n- Practice with diverse triangle side lengths to strengthen intuition.", "---", "### Conclusion", "Mastering equations like ( 5^2 + b^2 = 13^2 ) strengthens your ability to work with right triangles. Whether you're studying math, designing structures, or exploring spatial reasoning, the Pythagorean theorem offers a reliable tool for solving real-world problems. Keep practicing, and let geometry keep you ahead!", "---", "Keywords: Pythagorean theorem, ( 5^2 + b^2 = 13^2 ), solve triangles, right triangle geometry, squaring numbers, mathematical problem solving\nMeta Description: Learn how to solve ( 5^2 + b^2 = 13^2 ) using the Pythagorean theorem with step-by-step explanation and real-world applications. Perfect for math students and educators."]