History

The theorem is named after Pythagoras, but knowledge of right-triangle relationships existed in earlier Babylonian and Indian mathematics. It became central in Euclidean geometry and later in coordinate methods.

Derivation and reasoning

Area-based proof idea

A classic proof rearranges four identical right triangles inside a square in two different ways. Equating uncovered areas leads to \(a^2+b^2=c^2\).

Checkpoints

Find hypotenuse if legs are 9 and 12.

Answer: \(c=\sqrt{9^2+12^2}=\sqrt{225}=15\).

A right triangle has hypotenuse 17 and one leg 8. Find the other leg.

Answer: \(\sqrt{17^2-8^2}=\sqrt{225}=15\).

Are sides 10, 24, 26 right-angled?

Answer: Yes, \(10^2+24^2=100+576=676=26^2\).

Distance between \((-2,1)\) and \((4,9)\).

Answer: \(\sqrt{6^2+8^2}=10\).

Rectangle has side lengths 7 and 24. Find diagonal.

Answer: \(\sqrt{7^2+24^2}=25\).

Applications

• Architecture: checking right angles and structural diagonals.
• Navigation: shortest displacement calculations on grid maps.
• Computer graphics: pixel distance and vector magnitude.

References

• Standard high-school geometry curricula on Pythagoras and its converse.
• Classical Euclidean geometry references.