Pythagoras' Theorem

Pythagoras’ Theorem

Pythagoras’ theorem is a fundamental principle in geometry, specifically in triangle theory. Below are key points and formulas related to it.

Basic Principle

• Pythagoras’ theorem applies to every right-angled triangle. The theorem states that the square of the length of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides.
• It is mathematically represented in the formula: a² = b² + c², where a is the hypotenuse, and b and c are the other two sides.

Practical Applications

• Pythagoras’ theorem is used in numerous mathematical concepts, such as trigonometry, the calculation of distances (in various fields like astronomy, navigation, architecture etc.), the calculation of angles, and even in complex numbers.
• It can be used to find the length of any side in a right-angled triangle if the lengths of the other two sides are known.

Proofs

• Pythagoras’ theorem can be proved in numerous ways. One common proof is the geometric proof, which involves rearranging similar triangles so that a bigger square, composed of two smaller squares, is formed.
• Algebraic proofs, using aspect ratios of similar triangles, and reductio ad absurdum proofs can also be used.

Limitations of Pythagoras’ Theorem

• Keep in mind that Pythagoras’ theorem only works for right-angled triangles. Attempting to apply this to non-right angled triangles will result in inaccurate results.
• There exist formulas similar to Pythagoras’ theorem for non-right angled triangles, such as the law of cosines, which includes an additional term involving the cosine of the included angle.