To determine if the triangle is a right triangle, check if \(a^2 + b^2 = c^2\) where \(c\) is the longest side.

["# How to Determine If a Triangle Is a Right Triangle: The Pythagorean Theorem Explained", "Triangles are fundamental shapes in mathematics, and one of the most important features you can identify in any triangle is whether it is a right triangle. A right triangle is defined as a three-sided polygon with one angle measuring exactly 90 degrees. Recognizing a right triangle helps in solving geometric problems, calculating distances, and understanding trigonometric relationships. Fortunately, there is a simple and powerful method to determine if a triangle is right-angled: using the Pythagorean Theorem.", "## What Is the Pythagorean Theorem?", "The Pythagorean Theorem states that in a right 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 lengths of the other two sides. In algebraic terms, for sides (a), (b), and (c)—where (c) is the longest side (hypotenuse)—the theorem is written as:", "[\na^2 + b^2 = c^2\n]", "This relationship only holds true for right triangles. If this equation is satisfied, the triangle must be right-angled. If not, the triangle is acute or obtuse.", "## How to Apply the Theorem to Determine a Right Triangle", "Determining whether a triangle is right-angled using (a^2 + b^2 = c^2) is straightforward:", "1. Identify the longest side – Call this side (c). In any triangle, only one side can be the longest, and in a right triangle, this specific side is the hypotenuse.", "2. Measure or know the lengths of all three sides – You need the numerical values of sides (a), (b), and (c).", "3. Substitute values into the Pythagorean equation – Plug in the side lengths into the formula (a^2 + b^2 = c^2) and calculate both sides.", "4. Compare the results – If the left side equals the right side exactly, the triangle is a right triangle. If not, it is not right-angled.", "### Example:", "Consider triangle sides measuring 3, 4, and 5 units:", "- Let (a = 3), (b = 4), and (c = 5) (longest side).\n- Compute:\n (a^2 + b^2 = 3^2 + 4^2 = 9 + 16 = 25)\n (c^2 = 5^2 = 25)\n- Since (25 = 25), the triangle is a right triangle.", "---", "## A Quick Sample Table", "| Side (a) | Side (b) | Side (c) | Verify (a^2 + b^2 = c^2)? | Result |\n|------------|------------|------------|-------------------------------|--------------|\n| 5 | 12 | 13 | (25 + 144 = 169) → (169 = 169) | Right triangle |\n| 6 | 8 | 10 | (36 + 64 = 100) → (100 = 100) | Right triangle |\n| 7 | 10 | 15 | (49 + 100 = 149), (15^2 = 225)→Not equal | Not a right triangle |", "---", "## Why This Method Works", "The Pythagorean Theorem relies on fundamental Euclidean geometry, reflecting the properties of right-angled triangles in a two-dimensional plane. When the sum of the areas of two squares (on sides (a) and (b)) equals the area of a square (on side (c)), it confirms a right angle between sides (a) and (b).", "---", "## Practical Applications", "Knowing how to recognize a right triangle using this test is useful in:", "- Geometry proofs – Establishing right angles in problems\n- Architecture & engineering – Ensuring structural accuracy\n- Navigation & mapping – Computing distances using Pythagorean-based methods\n- Everyday problem-solving – From calculating roof slopes to planning garden layouts", "---", "## Summary", "To determine if a triangle is right-angled, use the Pythagorean identity:\n[\na^2 + b^2 = c^2\n]\nwhere (c) is the longest side. This simple algebraic check provides a powerful tool to confirm right triangles and unlock a world of geometric possibilities.", "Mastering this method empowers you to confidently identify right triangles and apply their properties across math, science, and real-world design."]