c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13

["Understanding the Pythagorean Theorem: Calculating the Hypotenuse of a Right Triangle", "When studying geometry, one of the most fundamental and powerful concepts is the Pythagorean Theorem—a cornerstone in triangle mathematics. Whether you're solving academic problems or applying geometry in real-world scenarios, knowing how to compute the hypotenuse of a right triangle using the formula ( c = \sqrt{a^2 + b^2} ) is essential.", "### What Is the Pythagorean Theorem?", "The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This relationship is written mathematically as:", "[\nc = \sqrt{a^2 + b^2}\n]", "where:\n- ( c ) represents the hypotenuse,\n- ( a ) and ( b ) are the lengths of the other two sides.", "### How to Apply the Formula", "Consider a right triangle with legs measuring 5 units and 12 units. To find the length of the hypotenuse, we substitute into the formula:", "[\nc = \sqrt{5^2 + 12^2}\n]", "First, calculate the squares:", "[\n5^2 = 25 \quad \ ext{and} \quad 12^2 = 144\n]", "Next, add these values:", "[\n25 + 144 = 169\n]", "Finally, take the square root:", "[\nc = \sqrt{169} = 13\n]", "Thus, the hypotenuse measures exactly 13 units.", "### Why This Matters in Real Life", "Beyond classroom geometry, the Pythagorean Theorem is widely used in fields such as architecture, construction, navigation, and computer graphics. For example, carpenters rely on it to ensure structures are square, while GPS systems use similar principles to calculate distances between points.", "### Quick Summary", "- Use ( c = \sqrt{a^2 + b^2} ) for right triangles.\n- Squaring the leg lengths eliminates square roots from addition.\n- The sum under the root simplifies neatly in this case: ( 25 + 144 = 169 ).\n- ( \sqrt{169} = 13 ), confirming the hypotenuse is 13 units.", "Mastering this simple formula unlocks a powerful algebraic and geometric tool, proving once again why the Pythagorean Theorem remains a vital part of mathematics education.", "---", "Keywords: Pythagorean Theorem, calculate hypotenuse, right triangle math, square root formula, geometry tutorial, a² + b² = c², understand hypotenuse, math problem solving."]