Pythagorean Theorem
Pythagorean's Theorem
Pythagorean's theorem is a fundamental principle in mathematics that establishes a relationship between the sides of a right-angled triangle. According to the theorem, the square of the length of the hypotenuse
(side c) is equal to the sum of the squares of the other two sides (a and b). This theorem finds its extensive application not only in geometry, where it aids in solving triangle-related problems, but also in various other fields of mathematics. In algebra, it enables us to work with distance formula and apply it to vectors and complex numbers. Given the x and y coordinates of a point, we can use this theorem to find the distance from the origin to a given point.
Play around with the applet below by dragging around the red point, move it up and down and side to side. Observe how all three sides of this triangle always follow the relation outline above. Observe how given the x and y coordinates of this point, the distance could always be found using this theorem. Try to find combinations of numbers for x, y, and c that are whole numbers. These triplets are called Pythagorean triplets.
To begin to get an intuitive understanding for why the theorem is true, let's begin to interpret the theorem geometrically. c, a, and b are all lengths of the sides of the sides of triangles. The geometric interpretation of their squaring would be a literal square. That means if we have three squares the sum of the area of and must equal the area of .
Play around with the triangle below, change the dimensions of its sides and change the orientation of the triangle. Observe how the sum of the areas and is always the same as the area of
The Theorem at Extremes
Play around with the triangle and report what happens at extremes. What happens when you make one side of the triangle, lets say side a, negligibly small compared to the other side? What happens when you make one side of the triangle, lets say side a, overwhelmingly large compared to the other side? Try to explain how the transition from one extreme to the other is predicted by the theorem.
The applet below outlines a proof of the Pythagorean Theorem. The white space in the center is equal to and the side of all the corner tringles are given by a and b, which we know are right triangles because they are constructed from the corners of a square. Moving the sliders rearranges the position of some of these corner triangles.
Logically the white spaces must still be the same area as they were before but with the new positions of the corner triangles we can see a new definition for this area. The lower left white space has sides a and a, and is clearly and the upper one is clearly . Their sum must still be the same as before so
Given the construction above and demonstration outlined by moving the triangles around write an algebraic proof that compliments this geometric proof. Hint: the area of must be the area of the larger square minus the area of the four corner triangles. Turn this statement into an expression and prove the Pythagorean Theorem.
This lesson has introduced what Pythagorean's theorem is, and some of it's many important uses. The first applet demonstrates one of it's more practical uses, to find the distance of a point, and allows one to get a feeling for what types of values the relation allows.
It is always important to get many different types of understandings of a theorem and the second applet literally translates the algebraic theorem into a visual representation. This is exactly what the theorem looks like and manipulating the tringles to extreme points should lead one to a better understanding of how the theorem behaves for all intermediary values.
One of the actual proofs of the theorem was outline in the third task. Given the visual and the geometric demonstration should be enough for one to connect the dots enough to be able to write their own algebraic proof of the theorem from the geometric construction they've been shown.