Understanding the Mathematical Subject the Pythagorean Theorem

Triangles have three connecting line segments called sides in plane geometry. The three line segments intersect at three vertices.  

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 two angles besides the right angle add up to 90 degrees. Examples are 45, 45, 90; 30, 60, 90; 20, 70, 90; etc. Two triangles have 2×180=360 degrees, which is the same number of degrees a circle has. 

The Pythagorean Theorem is only correct for right triangles.  The Pythagorean Theorem claims that the square of the hypotenuse is equal to the sum of the squares 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 a hypotenuse of 5 and two sides of 3 and 4.  It can be proven using similar triangles. It was also proven in Euclid’s Elements.

The Pythagorean Theorem is important in trigonometry.  The sine, cosine, tangent, co-tangent, secant, and co-secant can be found using the Pythagorean Theorem for right triangles.  The two angles of a right triangle that are not the right angle and the right angle are used.  The side opposite one of the angles can be called y.  The side adjacent to side y that is not opposite the right angle can be called x.  The side opposite the right angle z 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.

The third side of a right triangle can be found algebraically if the other two sides are known.  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 1/2 means 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, or irrational numbers.  In fact, the Pythagorean Theorem can be used to demonstrate what irrational numbers are.  It is known that if the two sides of a right triangle, neither of which is the hypotenuse, are each 1, then the hypotenuse c is c=(a^2+b^2)^1/2.  Since the two sides are both 1, 1^2+1^2=c^2, or 2=c^2.  The quantity c is found by taking the square root of each side of the equation.  The square root of 2 is 1.41421356237 to eleven decimal places.  The square root of 2 has been proven irrational.  Any square root of any integer that is not a perfect square is irrational.  Since the hypotenuse is an exact distance determined by the other sides, this proved that irrational numbers are on the number line.

Pascal’s Triangle is useful for many calculations including the binomial theorem, finding prime numbers, and powers of two.

The triangle inequality states that the sum of any two sides is longer than the third side.