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Triangles have three connecting line segments called sides in plane geometry. The Pythagorean Theorem was discovered by the School of Pythagoreans, led by Pythagoras of Ancient Greece. It is one of the most important theorems of mathematics. It involves right triangles, which are triangles that have one angle that is 90 degrees, called the hypotenuse. Since all triangles in plane geometry have 180 degrees, the other two angles add up to 90 degrees. Examples are 45, 45, 90; 30, 60, 90; 20, 70, 90; etc.
The Pythagorean Theorem only works for right triangles.
The Pythagorean Theorem claims that the square of the hypotenuse is equal to the square of the sum of the other two sides. This is a^2+b^2=c^2 in mathematical terms, where a, b are the two sides other than the hypotenuse, and c is the hypotenuse. An example is 3^2+4^2=5^2, or 9+16=25. This would be a right triangle with hypotenuse of 5 and two sides of 3 and 4.
If you know two sides of a right triangle, you can find the third by substituting the two values, then solving for the remaining variable. This means c=(a^2+b^2)^1/2, b=(c^2-a^2)^1/2, and a=(c^2-b^2)^1/2.
The exponent means take the square root of whatever is in the parenthesis.
The values for the three sides of the right triangle do not have to be integers. They can be integers, fractions, and imaginary numbers. In fact, the Pythagorean Theorem can be used to demonstrate what irrational numbers are as follows: If the two sides of a right triangle, neither of which is the hypotenuse,
are each 1, and the hypotenuse c is c=(a^2+b^2)^1/2. . Since the two sides are both 1, we can solve for c: 1^2+1^2=c^2, or 2=c^2. Now we solve for c by taking the square root of each side of the equation:
c=(2)^1/2. The square root of 2 is 1.41421356237 .
The Pythagorean Theorem is important in trigonometry. The sine, cosine, tangent, co-tangent, secant, and co-secant are found using the Pythagorean Theorem for right triangles. First, we use one of the
two angles of a right triangle that are not the hypotenuse (the right angle). The side opposite this angle can be called y. The side adjacent to the chosen angle that is not the hypotenuse can be called x. Let's
call the side opposite the right angle z (this is the hypotenuse. The definitions are sine=y/z, cosine=x/z,
tangent=y/x, cotangent=x/y, secant=z/x, and cosecant=z/y.

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