Pythagorean Theorem, Statistics Problem Example

Pythagorean Theorem is a mathematical statement which serves as a fundamental rule in Euclidean geometry. This theorem relates the three sides of a right angled triangle. Mathematically it states that the square of the longer side of the right angled triangle (hypotenuse) is equal to the sum of the squares of the shorter sides of the right angled triangle that meet at right angles. Or it can be defined by using areas that; for any right angled triangle, the area of the square whose side is opposite the right angle is equal to the areas of the squares whose sides meet at a right angle.

The theorem can be represented as an equation relating the lengths of the sides a, b, and c of a right triangle often referred to as the Pythagoras equation

Where c is the hypotenuse a is the base and b is the height of the right triangle.

This theorem is named after the Greek mathematician called Pythagoras

Proof of theorem using similar triangles

Proof using similar triangles lies on the proportionality of the two sides of two similar triangles. From geometry the ratio of two corresponding sides of the triangle is the same regardless of the size of the triangles.

Let XYZ represent a right angled triangle with the right angle located at Z as per the figure above. By drawing a line from point Z it meets line XY perpendicular at point H. the point H divides the length of the hypotenuse into two parts b and c. now we have two triangles which are similar that is triangle XYZ and XZH. They are similar since they both have aright angle and share the angle at X implying that their third angle represented as ? in the figure will be the same. In the same reasoning triangle ZYH is similar to XYZ. By triangle postulate the sum of the angles in a triangle is 180⁰ and is equivalent to the pararell postulate.

Thus by the similarity of the triangles the ratio of the sides can be equated as follows

From the above result the first part equates the  cosine of angle ? and the second equates the sines.

The above ratios can be written as

And

Adding the two equations we get

Thus by rearranging we have

Which is Pythagoras theorem.

Works cited

Heath, Thomas.” The Theorem of Pythagoras” A History of Greek Mathematics 2 VolsClarendon Press, Oxford. p. 144 .1921.