Pythagorean Theorem — a² + b² = c²
A right triangle has one 90° angle. The Pythagorean theorem (a² + b² = c²) relates its three sides: legs a and b, and hypotenuse c (opposite the right angle). Right triangles are fundamental in trigonometry, construction, and navigation.
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Tip: To check if a triangle is a right triangle: compute a²+b² and c². If equal, it's a right triangle. Also: if a triangle's longest side satisfies c²>a²+b², it's obtuse; if c²<a²+b², it's acute.
- 1Hypotenuse: c = √(a² + b²)
- 2Leg: a = √(c² − b²)
- 3Angles: angle A = arctan(a/b), angle B = arctan(b/a), sum of angles = 180°
- 4Area = (1/2) × base × height = (1/2) × a × b
Legs 3 and 4=Hypotenuse = 5, angles 36.87° and 53.13°Classic 3-4-5 right triangle
Hypotenuse 10, one leg 6=Other leg = 8√(100−36) = √64 = 8
| a | b | c | Multiple |
|---|---|---|---|
| 3 | 4 | 5 | Base triple |
| 5 | 12 | 13 | Base triple |
| 8 | 15 | 17 | Base triple |
| 6 | 8 | 10 | 2× (3,4,5) |
| 9 | 12 | 15 | 3× (3,4,5) |
| 7 | 24 | 25 | Base triple |
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Fun Fact
The 3-4-5 right triangle was known to ancient Egyptians, who used knotted ropes with 12 equal spaces to create right angles for surveying land after Nile floods. This "rope-stretching" method is one of the earliest practical applications of the Pythagorean theorem.
References
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