Solve a² + b² = c² for missing hypotenuse or leg with step-by-step radical simplification and exact decimal solutions.
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The Pythagorean theorem states that in any right-angled triangle, the area of the square whose side is the hypotenuse (c) is equal to the sum of the areas of the squares on the other two sides (legs a and b): a² + b² = c².
c = \sqrt{a^2 + b^2}, \quad a = \sqrt{c^2 - b^2}, \quad b = \sqrt{c^2 - a^2}To find hypotenuse c, sum the squares of the legs and take the square root. To find a leg, subtract the known leg squared from hypotenuse squared and take the root.
Scenario: Find hypotenuse c for legs a = 9 and b = 12
Apply theorem: c² = a² + b²
Substitute values: c² = 9² + 12² = 81 + 144 = 225
Solve for c: c = √225 = 15
No. The Pythagorean theorem only applies to right triangles (90° angle). For general triangles, use the Law of Cosines (c² = a² + b² - 2ab cos C).
Attributed to ancient Greek mathematician Pythagoras of Samos (c. 570–495 BC), though Babylonian and Indian mathematicians knew the relationship centuries earlier.
A Pythagorean triple consists of three positive integers (a, b, c) such that a² + b² = c², such as (3, 4, 5) and (5, 12, 13).
Factor out perfect squares from under the square root. For example, √72 = √(36 × 2) = 6√2.
Solve right triangles with hypotenuse, legs, area, perimeter, altitude, and acute angles using trigonometry and the Pythagorean theorem.
Calculate triangle area using Heron’s formula or base and height, perimeter, semi-perimeter, inradius, and classification.
Calculate the square root of positive and negative numbers (√x). Provides radical simplification, perfect square check, and imaginary roots.