Thanks cheeky Andy who thought that last week’s email was so loopy he asked “….…have you been sniffing those permanent markers again?!” cuh!  I thought it was perfectly sensible and understandable.  So this week, I will not make any jokes at all for fear of being chastised, so I am going to tell you about some maths

Pythagorean Theorem

The theorem is as follows:

In any right triangle, the area of the square whose side is the hypotenuse (the side of a right triangle opposite the right angle) is equal to the sum of areas of the squares whose sides are the two legs (I.e. the two sides other than the hypotenuse).

If we let c be the length of the hypotenuse and a and b be the lengths of the other two sides, the theorem can be expressed as the following equation:

This theorem may have more known proofs than any other. The Pythagorean Proposition, a book published in 1940, contains 370 proofs of Pythagoras’ theorem, including one by American President James Garfield

Like many of the proofs of the Pythagorean theorem, this one is based on the proportionality of the sides of two similar triangles.

Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. We draw the altitude from point C, and call H its intersection with the side AB. The new triangle ACH is similar to our triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well. By a similar reasoning, the triangle CBH is also similar to ABC. The similarities lead to the two ratios:

These can be written as:

and

Summing these two equalities, we obtain:

In other words, the Pythagorean theorem:

That’ll teach you ha HA.

Trackback URL