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Math and Arithmetic

What is the Pythagorean theorem?


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November 10, 2017 10:58AM

Since the 4th century AD, Pythagoras has commonly been given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right triangle the square of the hypotenuse (the longest side, opposite the right angle) is equal to the sum of the squares of the other two sides.


In other words, given a right-triangle with side lengths a and b, and a hypotenuse of length c, then

a2 + b2 = c2


Given a right-triangle with side a = 3 and side b = 4, what is the hypotenuse (side c)?

Solution: a2 + b2 = c2

32 + 42 = c2

(3 x 3) + (4 x 4) = c2

9 + 16 = c2

25 = c2

c = √25

c = 5.

(Note that c can't equal -5 because c is the length of the hypotenuse of the triangle and length must be positive.)

The importance of the theorem goes way beyond triangles, in fact the Pythagorean theorem is the basis for the definition of distance between two points in space of any dimension of size 2 or more.

(There is a related link to 81 different proofs of this theorem.)

Euler Improvement

The mathematician George Euler improved the Pythagoras theorem to apply to all triangles using the cosine of the included angle:

a2 + b2 -2abcosT= c2

where T is the angle between a and b and cos the goniometric function.

(The cosine of 90° is 0 which makes this the Pythagoras theorem.)


IF BC=A=5CM=base of right angle, and AB=B=6CM the perpendicular and AC=C=the hypotenuse.



So we have:



Now we use the square root property but take the positive square root.

So C is approximately equal to 7.81 CM

Generalizing the theorem to higher dimensions

The Pythagorean Theorem works in higher dimensions too. If you have three legs, each one in a different dimension, and each at right angles to the other two, the hypotenuse joining these three lines has a length which equals sqrt(a2+b2+c2). You can't have four mutually right legs in three dimensions, but you can in four dimensions, in which case h=sqrt(a2+b2+c2+d2) and so on.
a squared + b squared = c squared. You square a and square b, and them together and find the square root of that number. that = c
A2 + B2 = C2