A right triangle has one right angle (90°) and two minor angles (<90°).
The Pythagorean Theorem states that:
Geometrically, it can be verified that in any right triangle it is true that the sum of the areas of the squares formed on its legs is equal to the area of the square built on its hypotenuse, that is:
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Note. Courtesy of the author: José María Pareja Marcano. Chemist. Seville, Spain.
The Converse of the Pythagorean Theorem
- If c2 = a2 + b2 then is a right triangle because the Pythagorean Theorem is verified.
- If c2 < a2 + b2 then is an acute triangle because the angle facing side c is an acute angle. This inequality applies for all sides: b2 < a2+c2 and a2 < b2+c2.
- If c2 > a2 + b2 then is an obtuse triangle. The square of the side opposite the obtuse angle is greater than the sum of the squares of the other two sides.
Can the Pythagoras theorem be applied on all triangles?
Let p the segment of the projection of side b onto side c. The formula of the generalized Pythagorean theorem is:
Segment p can change sign depending on the type of triangle:
- α=90°: projection on c p=0. The Pythagorean theorem is applied when the term is canceled. It’s a right triangle.
- α<90°: projection on c p>0. It’s an acute triangle.
- α>90°: projection on c p<0. It’s an obtuse triangle.
This form of the generalized Pythagorean theorem, expressed trigonometrically, is the law of cosines or cosine rule. The law of cosines states:
The 3D Pythagorean Theorem
The Pythagorean theorem can be extended to three-dimensional space by applying it to the length of the longest diagonal of a rectangular prism (or cuboid). This diagonal is the hypotenuse of a right triangle having one side on the edge of the prism and the other along a diagonal of a rectangular face of the prism.
Relationship between legs and hypotenuse
AUTHOR: Bernat Requena Serra