\[ c^2 = a^2 + b^2 = 16 + 9 = 25 \]

["# Understanding ( c^2 = a^2 + b^2 = 16 + 9 = 25 ): The Core of the Pythagorean Theorem", "Mathematics is full of elegant formulas that unlock powerful relationships, and one of the most celebrated is the Pythagorean Theorem. Meaningfully expressed as ( c^2 = a^2 + b^2 ), this equation forms the foundation of right triangle geometry and has countless practical applications in architecture, navigation, physics, and beyond.", "## 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, denoted as ( c )) is equal to the sum of the squares of the lengths of the other two sides (( a ) and ( b )):\n[\nc^2 = a^2 + b^2\n]", "When ( a = 3 ) and ( b = 4 ), plugging into the formula yields:\n[\nc^2 = 3^2 + 4^2 = 9 + 16 = 25\n]\nTaking the square root gives the hypotenuse:\n[\nc = \sqrt{25} = 5\n]", "Thus, the classic 3-4-5 triangle emerges, one of the simplest and most recognizable right triangles.", "## Calculating Step-by-Step", "Let’s break down the math:\n- ( a^2 = 3^2 = 9 )\n- ( b^2 = 4^2 = 16 )\n- Sum: ( 9 + 16 = 25 )\n- Therefore, ( c^2 = 25 \Rightarrow c = 5 )", "This straightforward calculation validates the Pythagorean identity for these side lengths.", "## Why Is This Formula Important?", "Understanding ( c^2 = a^2 + b^2 ) allows you to:", "- Determine missing side lengths in a right triangle.\n- Verify whether a triangle is right-angled (by checking if ( a^2 + b^2 = c^2 )).\n- Solve real-world problems involving distance, height, or diagonal measurements.", "## Applications of the Pythagorean Theorem", "- Construction & Architecture: Ensuring walls, floors, and roofs meet at precise right angles.\n- Navigation & GPS: Calculating straight-line distances between two points.\n- Physics: Breaking vectors into components or computing resultant velocities.\n- Computer Graphics: Calculating pixel distances in 2D spaces.", "## Exploring More: Pythagorean Triples", "The 3-4-5 triangle is just one example of Pythagorean triples—sets of integers ( (a, b, c) ) that satisfy ( a^2 + b^2 = c^2 ). Other well-known triples include:\n- ( 5-12-13 )\n- ( 7-24-25 )\n- ( 8-15-17 )", "These triples illustrate natural number solutions to the theorem and have applications in number theory and cryptography.", "## Final Thoughts", "The formula ( c^2 = a^2 + b^2 = 16 + 9 = 25 ) elegantly demonstrates the Pythagorean Theorem’s simplicity and power. Whether you're measuring a roof slope, plotting a course, or exploring number sets, this foundational equation empowers precise calculation and deeper geometric insight.", "Mastering ( a^2 + b^2 = c^2 ) not only strengthens your understanding of geometry—it opens doors to limitless problem-solving opportunities across science, engineering, and daily life.", "---", "Keywords:\nPythagorean Theorem, ( c^2 = a^2 + b^2 ), 3-4-5 triangle, right triangle, Pythagorean triples, geometry, math formula, distance calculation, trigonometry basics, constructive geometry", "Meta Description:\nDiscover how ( c^2 = a^2 + b^2 = 16 + 9 = 25 ) illustrates the Pythagorean Theorem — the essential square relationship in right triangles, with real-world uses and key insights for students and professionals alike."]