Pythagorean Theorem
A triangle is composed of three line segments. When looking at a right triangle, it is composed of two legs and one hypotenuse. The hypotenuse is the length of the triangle which is the longest and is also opposite to the right angle. There is a theorem, called the Pythagorean Theorem, which states that in every right triangle, a2 + b2 = c2. Where “a” and “b” are the two legs and “c” is the hypotenuse. The theorem is stating that the square of the hypotenuse is equal to the sum of the squares of each of the two legs of the triangle.
Pythagoras' Proof
Below is a right triangle, CBA. Let us conclude that segments CA and AB are both the legs of the triangle and CB is the hypotenuse of the triangle since it is opposite the right angle (depicted by the square). When moving the slider, “Translation”, we can see that the pieces of each leg translate construct a perfect square over the hypotenuse. This shows that even though the hypotenuse is the largest side, the theorem still holds: a2 + b2 = c2.
Pythagoras Fractal Tree
The Pythagorean Theorem can be shown in numerous visual ways. Another way to show the Pythagorean Theorem is through the Pythagoras Fractal Tree. The fractal tree is essentially composed of similar set up like the above applet, where there are squares set up in triplets with a triangle forming in the middle. The key component of the Pythagoras Fractal Tree is the fact that it is a fractal.
A fractal is a geometrical figure in which each part has the same characteristics of the whole figure. Fractals are typically in recurring patterns where the objects progressively become smaller. To apply to everyday life, think of a snowflake. A snowflake has a specific pattern on it all-around where it starts off large and progressively becomes smaller as it reaches the outer edges of the snowflake. This is the prime, and most known example of a fractal-like pattern.
Looking below at the applet, you are able to see a set up of a Pythagoras Fractal Tree. The tree has been defaulted to having a 90 degree angle with the base triangle which reflects all of the triangles in tree to have the same angle measurement. When you adjust the slider, “Slide Me!”, you will be able to see the Pythagorean Theorem occur in a different visual way, yet again. This time, you are able to see the tree form a straight line with the hypotenuses of each set of triplet due to minimizing or maximizing of the angle size.