Use the Pythagorean theorem: \( a^2 + b^2 = c^2 \) for a right triangle.

["# Use the Pythagorean Theorem: ( a^2 + b^2 = c^2 ) for a Right Triangle", "The Pythagorean theorem is one of the most fundamental principles in geometry, forming the cornerstone of understanding right triangles. Discovered by the ancient Greek mathematician Pythagoras, this elegant equation — ( a^2 + b^2 = c^2 ) — relates the lengths of the sides of a right-angled triangle. Whether you’re studying math, designing structures, or solving real-world problems, using the Pythagorean theorem to find unknown side lengths is a powerful skill.", "## What Is a Right Triangle?", "A right triangle is a triangle with one angle exactly equal to 90 degrees. The side opposite the right angle is called the hypotenuse, the longest side, denoted as ( c ). The other two sides, which meet at the right angle, are called the legs, labeled ( a ) and ( b ).", "## How the Pythagorean Theorem Works", "The Pythagorean theorem states that in any right triangle:", "[\na^2 + b^2 = c^2\n]", "Where:\n- ( a ) and ( b ) are the lengths of the legs\n- ( c ) is the length of the hypotenuse", "Why does it work?\nThe theorem reflects a profound relationship between area and distance in geometry. By squaring each side and summing the squares of the legs, we calculate the equal combined area, equal to the square of the hypotenuse — an expression of geometric consistency and symmetry.", "## Solving for Unknown Sides", "Using the Pythagorean theorem to find an unknown side is straightforward:", "### 1. Find the hypotenuse ( c )\nIf you know the lengths of the legs ( a ) and ( b ), rearrange the equation:\n[\nc = \sqrt{a^2 + b^2}\n]", "Example:\nSuppose ( a = 3 ) units and ( b = 4 ) units.\nThen,\n[\nc = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \ ext{ units}\n]", "### 2. Find a leg when hypotenuse and the other leg are known\nIf you know ( c ) and one leg, say ( b ), solve for ( a ):\n[\na = \sqrt{c^2 - b^2}\n]", "Example:\nIf ( c = 10 ) and ( b = 6 ), then\n[\na = \sqrt{100 - 36} = \sqrt{64} = 8\n]", "## Applications of the Pythagorean Theorem", "- Construction and architecture: Ensuring corners are square and framing is stable.\n- Navigation: Calculating shortest distances or vector displacements.\n- Computer graphics: Determining pixel distances and rendering 3D space.\n- Physics: Analyzing forces and vector components.\n- Everyday problem solving: From calculating screen sizes to estimating rooftop slopes.", "## Summary", "The Pythagorean theorem ( a^2 + b^2 = c^2 ) is more than an abstract formula — it’s a practical tool for solving geometry problems involving right triangles. Mastery of this theorem empowers you to find missing side lengths, verify right angles, and apply geometric principles across science, engineering, and daily life.", "Start using ( a^2 + b^2 = c^2 ) today to unlock a clearer understanding of right triangle relationships — and build your confidence in geometry with every calculation.", "---", "Keywords: Pythagorean theorem, right triangle, ( a^2 + b^2 = c^2 ), solve right triangle, geometry formulas, triangle geometry, vector mathematics, construction math."]