Since the sum of the squares of the two shorter sides equals the square of the longest side, it is a right triangle.
["Understanding Right Triangles: When the Sum of Squares Defines the Triangle", "When studying geometry, one of the most fundamental and elegant principles is the Pythagorean Theorem — a cornerstone law that defines right triangles. Since the sum of the squares of the two shorter sides equals the square of the longest side, this condition serves as the definitive test for identifying a right triangle.", "### What Is a Right Triangle?", "A right triangle is defined by having one internal angle that measures exactly 90 degrees. The side opposite this right angle is called the hypotenuse — the longest side of the triangle. The other two sides, known as the legs or shorter sides, play a critical role in determining whether the triangle satisfies the Pythagorean Theorem.", "### The Pythagorean Theorem Explained", "At the heart of a right triangle’s identity is the Pythagorean Theorem:\nIf a triangle has sides of lengths a, b, and c, where c is the hypotenuse, then\n[\na^2 + b^2 = c^2\n]\nThis equation holds true only for right triangles. If the sum of the squares of the two shorter sides is equal to the square of the longest side, then the triangle is confirmed to be right-angled.", "### How to Apply the Theorem", "To verify if a triangle is a right triangle:", "1. Identify the longest side — this is the potential hypotenuse.\n2. Square the lengths of the two shorter sides.\n3. Add them together.\n4. Compare the sum to the square of the longest side.", "If the equality holds, the triangle is a right triangle.\nIf not, it is an acute or obtuse triangle — not right.", "### Examples and Applications", "Consider a triangle with sides 3, 4, and 5.\nCalculate:\n( 3^2 + 4^2 = 9 + 16 = 25 )\n( 5^2 = 25 )\nSince ( 3^2 + 4^2 = 5^2 ), this confirms a right triangle with legs 3 and 4, and hypotenuse 5. This classic 3-4-5 triangle is widely used in construction, navigation, and education.", "Beyond theory, recognizing right triangles helps solve real-world problems — from ensuring structural stability to programming graphics in video games and designing sustainable architecture.", "### Why This Rule Matters", "Understanding that a right triangle is defined by the relationship ( a^2 + b^2 = c^2 ) transforms geometry from abstract shapes into practical, measurable tools. Whether you're an architect, student, or curious learner, this simple yet powerful criterion serves as a critical gateway to mastering geometric principles.", "Key Takeaway:\nA triangle is a right triangle if and only if the sum of the squares of the two shorter sides equals the square of the longest side. This timeless rule ensures clarity, precision, and confidence in identifying one of the most essential forms in geometry.", "---", "Explore more about Pythagorean theorem applications, solve interactive geometry problems, or dive into triangle properties with recommended geometry resources today!"]
