Pythagoras' Theorem

1. If a squared plus b squared is more than c squared, the triangle is acute. If c squared is more than a squared plus b squared, the triangle is obtuse.

2. Converse

2.1. Converse of Pythagoras' Theorem

3. What is it?

3.1. In mathematics, the Pythagorean theorem, also known as Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

4. Proof

4.1. Pythagoras' Theorem

4.2. "Let a, b, c denote the legs and the hypotenuse of the given right triangle, and consider the two squares in the accompanying figure, each having a+b as its side. The first square is dissected into six pieces-namely, the two squares on the legs and four right triangles congruent to the given triangle. The second square is dissected into five pieces-namely, the square on the hypotenuse and four right triangles congruent to the given triangle. By subtracting equals from equals, it now follows that the square on the hypotenuse is equal to the sum of the squares on the legs" Consider the following figure. The area of the first square is given by (a+b)^2 or 4(1/2ab)+ a^2 + b^2. The area of the second square is given by (a+b)^2 or 4(1/2ab) + c^2. Since the squares have equal areas we can set them equal to another and subtract equals. The case (a+b)^2=(a+b)^2 is not interesting. Let's do the other case. 4(1/2ab) + a^2 + b^2 = 4(1/2ab)+ c^2 Subtracting equals from both sides we have

4.3. The first thing to notice is that there are three three similar triangles in this picture. Why? Because: 1) similar triangles all have the same internal angles. https://stephenwhitt.files.wordpress.com/2009/02/pythag.gif?w=490https://stephenwhitt.files.wordpress.com/2009/02/pythag.gif?w=490 2) all triangles have the same internal angle measure (we know it’s 180 degrees, but that doesn’t matter – what matters is that all triangles have the same measure, whatever it is). 3) by 1 and 2, if two right triangles share an angle, then they have to be similar. The three similar triangles are: ABC, ACX, and CBX (notice that the order of the letters is crucial here, matching the appropriate angles). In similar triangles, the ratios of the sides are equal. So the triangle CBX compared to the triangle ABC gives a/x = c/a Cross-multiplying gives a^2 = cx The triangle ABC compared to the triangle ACX gives c/b = b/(c – x) Cross-multiply again and you get b^2 = c^2 – cx But we already know cx = a^2 from before, so now we have b^2 = c^2 – a^2 Rearranging, we get the familiar a^2 + b^2 = c^2

5. Corollary

6. Applications

6.1. Calculate the shortest path to a destination

6.2. Determine how tall a ladder must be to successfully and safely get into one or paint a building

7. History

7.1. The Pythagorean theorem takes its name from the ancient Greek mathematician Pythagoras (500 B.C.-569 B.C.), who was the first to offer a proof of the theorem.

7.1.1. Pythagoras was born on the island of Samos in Greece, and did much traveling through Egypt, learning, among other things, mathematics. Pythagoras gained his famous status by founding a group, the Brotherhood of Pythagoreans, which was devoted to the study of mathematics. The group was almost cult-like in that it had symbols, rituals and prayers. They also discovered irrational numbers.

7.2. However, even though Pythagoras is credited as the "father" of Pythagoras' theorem, Pythagoras wasn't the only one to find out about the relationship between the sides of a right-angled triangle. The others who may have found out and documented it were:

7.2.1. The Babylonian people

7.2.2. The Chinese, in an astronomical and mathematical treatise called Chou Pei Suan Ching

7.2.3. Ancient Indian mathematicians

7.2.4. Ancient Mayans