Use the Pythagorean theorem: \( a^2 + b^2 = c^2 \), where \( c \) is the longest side.

["# Mastering the Pythagorean Theorem: ( a^2 + b^2 = c^2 )", "The Pythagorean Theorem is one of the most foundational and widely used concepts in geometry, serving as a cornerstone for understanding right triangles. Its elegant formula—( a^2 + b^2 = c^2 )—lets you calculate the length of any side of a right triangle when the other two are known. Whether you're solving classroom problems, designing structures, or exploring real-world applications, understanding and applying this theorem is essential.", "## What is the Pythagorean Theorem?", "Formulated by the ancient Greek mathematician Pythagoras, the theorem applies only to right triangles—triangles with one 90-degree angle. The formula states that in such a triangle, the square of the hypotenuse (the side opposite the right angle, typically labeled ( c )) is equal to the sum of the squares of the other two sides (( a ) and ( b )). This simple yet powerful relationship forms the basis for countless mathematical and engineering principles.", "## How to Use the Pythagorean Theorem in Real Life", "### 1. Finding Missing Side Lengths\nIf you know the lengths of the two shorter sides (( a ) and ( b )), you can find the hypotenuse (( c )) using:\n[ c = \sqrt{a^2 + b^2} ]\nFor example, if ( a = 3 ) and ( b = 4 ), then ( c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 ).", "### 2. Determining Right Angles without Measuring\nThe theorem can verify whether a triangle has a right angle. Given side lengths ( a ), ( b ), and ( c ) (with ( c ) assumed the longest), calculate ( a^2 + b^2 ) and compare it to ( c^2 ):\n- If ( a^2 + b^2 = c^2 ): the triangle is right-angled.\n- If ( a^2 + b^2 > c^2 ): the triangle is obtuse.\n- If ( a^2 + b^2 < c^2 ): the triangle is acute.", "This is valuable in fields like architecture and navigation, where confirming perpendicularity ensures structural integrity and accuracy.", "### 3. Distance Calculations in Coordinate Geometry\nIn the Cartesian plane, the theorem underpins the distance formula:\n[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]\nHere, ( d ) represents the straight-line distance between two points, calculated using the same ( a^2 + b^2 = c^2 ) logic.", "## Why Understanding the Pythagorean Theorem Matters", "Beyond academic exercises, mastering this theorem strengthens spatial reasoning and problem-solving skills. It supports learning in trigonometry, engineering, physics, and computer graphics, where spatial relationships and distances are critical. More than a formula, the Pythagorean Theorem reveals a deep, inherent mathematical order in right-angled triangles that shapes how we navigate and design the world.", "## Final Thoughts", "Using the Pythagorean theorem ( a^2 + b^2 = c^2 ) isn’t just about solving equations—it’s about uncovering the geometric harmony between shapes. Whether you’re a student tackling homework, a builder laying a foundation, or an engineer plotting the next trajectory, this timeless principle remains an indispensable tool. Embrace it, and unlock a clearer, more accurate understanding of space."]