Inverse Pythagorean theorem

The converse of the Pythagorean theorem: if a triangle’s side lengths satisfy a² + b² = c², its angle opposite c is right.

Pythagorean theorem: In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs. Its side-length equation becomes a test for whether a triangle has a right angle.
Law of cosines: A formula relating a triangle’s side lengths to the cosine of one of its angles. Setting a² + b² = c² in this formula forces the opposite angle’s cosine to zero.
Surveying: The measurement and mapping of land boundaries, positions, and elevations. Surveyors can check whether measured lengths establish a right angle in a layout.
Pythagorean theorem converse error: A common reasoning mistake that treats the Pythagorean theorem’s implication as reversible without checking its conditions. The converse is valid for the sides of an actual triangle, not arbitrary lengths lacking a triangle.
Triangle: A polygon with three sides, three vertices, and three interior angles. The converse applies to triangles whose three side lengths are known.
Cosine: A trigonometric function that relates an angle in a right triangle to the ratio of adjacent side to hypotenuse. A zero cosine identifies the angle as 90 degrees in the law-of-cosines proof.
Construction layout: The transfer of design dimensions and positions onto a building site. A 3-4-5 triangle provides a direct field method for setting perpendicular lines.
Acute triangle: A triangle whose three interior angles are all less than 90 degrees. For the longest side c, a² + b² greater than c² indicates an acute opposite angle.
Right angle: An angle measuring 90 degrees, formed by perpendicular lines or rays. The theorem identifies the angle opposite the longest side as exactly 90 degrees.
Side-angle relationship in triangles: The geometric relationship in which larger angles face longer sides within a triangle. It helps identify c as the side opposite the angle being tested when c is longest.

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Geometric mean theoremCompared with: It tests whether a triangle is right, rather than relating an altitude to hypotenuse segments.