📙 Pythagorean Theorem

For a right triangle with legs $a$, $b$, and hypotenuse $c$:

\color{#5ff}
⌳    
\boxed{𝑎²+𝑏²=𝑐²}

The sum of the squares of the two shorter sides is equal to the square of the longest side: the hypotenuse.

This theorem is foundational in trigonometry, physics, and engineering, enabling calculations of distances and angles in a variety of fields.

🔰 Visual Proof

This theorem holds true not just for squares but for any similar shapes on the sides of a right triangle.

⚜️ Pythagorean Triples

A Pythagorean triple is a set of three whole numbers $(a, b, c)$ that satisfy the Pythagorean theorem.

a	b	c
3	4	5
5	12	13
7	24	25
8	15	17
9	40	41

These triples are useful in various geometric problems, computer graphics, and even cryptography.

⚜️ Applications of the Pythagorean Theorem

The theorem has numerous applications:

• Navigation and Distance Measurement: Used to calculate shortest paths.

• Architecture and Construction: Ensures right angles and structural integrity.

• Computer Graphics: Helps in rendering 3D objects and collision detection.

• Astronomy and Physics: Used to compute distances in space.

Understanding and applying the Pythagorean Theorem unlocks solutions to problems in a wide array of scientific and engineering disciplines.