In geometry, a theorem stating that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. If the hypotenuse is h units long and the lengths of the other sides are a and b, then h2 = a2 + b2.
The theorem provides a way of calculating the length of any side of a right-angled triangle if the lengths of the other two sides are known. For example, to find the length of a bridge constructed over a valley 120 m wide, when the vertical drop of the bridge is 20 m:
Using Pythagoras' theorem B2 = 1202 + 202, so B2 = 14,400 + 400 = 14,800, giving b = √14,800 = 121.66 (to two decimal places).
The length of the bridge is 121.66 m.
Pythagoras' theorem is also used to determine certain trigonometric identities such as sin2 θ + cos2 θ = 1.
The Origins of Mathematics
Pythagoras' theorem: working out angles
At the Bottom of the Garden – Practical Use of Pythagoras' Theorem
It's That Man Again – Pythagoras and His Theorem
Dave's Short Trig Course
SOS Mathematics: Trigonometry
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