Question

what is Pythagorean Theorem.

Answer 1

The Pythagorean Theorem is a fundamental relation in geometry that applies to right triangles (triangles with one 90° angle).

Statement
- If a and b are the lengths of the two legs (the sides that meet at the right angle) and c is the length of the hypotenuse (the side opposite the right angle), then
a^2 + b^2 = c^2.

Example
- For a triangle with legs 3 and 4, the hypotenuse is c = sqrt(3^2 + 4^2) = sqrt(9 + 16) = 5. (This gives the Pythagorean triple 3–4–5.)

Converse
- If for three positive numbers a, b, c we have a^2 + b^2 = c^2, then a triangle with side lengths a, b, c is a right triangle (with c as the hypotenuse).

Uses and connections
- Computing distances in the plane: distance between points (x1,y1) and (x2,y2) is sqrt((x2−x1)^2 + (y2−y1)^2), which comes from the theorem.
- Pythagorean triples: integer solutions like (3,4,5), (5,12,13).
- Generalization: the law of cosines extends it to non-right triangles.

Proof idea (brief)
- One common proof uses similar triangles: dropping an altitude from the right angle creates smaller triangles similar to the original; comparing their side ratios leads to a^2 + b^2 = c^2.
- Another visual proof rearranges four copies of the triangle inside a square to show the equality of areas.

If you want, I can show a step-by-step proof, more examples, or how to use it in coordinate geometry.