Though similar concepts had been discovered by the Babylonians, Greek Mathematician Pythagoras was the first person to come up with a geometric proof about how the sum of the squares of the lengths can determine the side lengths of a right triangle.
Pythagoras determined that when three squares are arranged so that they form a right angle triangle, the largest of the three squares must have the same area as the other two squares combined.
In the picture below, you can see how the sum of the squares creates the right triangle ABC. Squares are different from other parallelograms and trapezoids because all their sides are equal lengths. So since squares are made up of four equal sides, you can see that each individual square makes up a side of the right triangle.
The length of the largest square, which we'll call length c , is the length of the hypotenuse. The hypotenuse is the longest side of a right triangle. You can use it and two lengths to find the shortest distance. For instance, if you are at sea and navigating to a point that is miles north and miles west, you can use the theorem to find the distance from your ship to that point and calculate how many degrees to the west of north you would need to follow to reach that point.
The distances north and west will be the two legs of the triangle, and the shortest line connecting them will be the diagonal. The same principles can be used for air navigation.
For instance, a plane can use its height above the ground and its distance from the destination airport to find the correct place to begin a descent to that airport. Surveying is the process by which cartographers calculate the numerical distances and heights between different points before creating a map. Because terrain is often uneven, surveyors must find ways to take measurements of distance in a systematic way. The Pythagorean Theorem is used to calculate the steepness of slopes of hills or mountains.
A surveyor looks through a telescope toward a measuring stick a fixed distance away, so that the telescope's line of sight and the measuring stick form a right angle. Since the surveyor knows both the height of the measuring stick and the horizontal distance of the stick from the telescope, he can then use the theorem to find the length of the slope that covers that distance, and from that length, determine how steep it is.
Jon Zamboni began writing professionally in He has previously written for The Spiritual Herald, an urban health care and religious issues newspaper based in New York City, and online music magazine eBurban. How could the small triangles not add to the larger one? Actually, it turns out the Pythagorean Theorem depends on the assumptions of Euclidean geometry and doesn't work on spheres or globes, for example. But we'll save that discussion for another time.
We used triangles in our diagram, the simplest 2-D shape. But the line segment can belong to any shape. Take circles, for example:. Pretty wild, eh? We can multiply the Pythagorean Theorem by our area factor pi, in this case and come up with a relationship for any shape. Remember, the line segment can be any portion of the shape. We could have picked the circle's radius, diameter, or circumference -- there would be a different area factor, but the relationship would still hold.
So, whether you're adding up pizzas or Richard Nixon masks, the Pythagorean theorem helps you relate the areas of any similar shapes. Now that's something they didn't teach you in grade school. The Pythagorean Theorem applies to any equation that has a square. In reality, the "length" of a side can be distance, energy, work, time, or even people in a social network:. Metcalfe's Law if you believe it says the value of a network is about n 2 the number of relationships.
In terms of value,. Pretty amazing -- the 2nd and 3rd networks have 70M people total, but they aren't a coherent whole. The network with 50 million people is as valuable as the others combined. Some programs with n inputs take n 2 time to run bubble sort, for example. In terms of processing time:. Pretty interesting. Given this relationship, it makes sense to partition elements into separate groups and then sort the subgroups. Indeed, that's the approach used in quicksort, one of the best general-purpose sorting methods.
The Pythagorean theorem helps show how sorting 50 combined elements can be as slow as sorting 30 and 40 separate ones. We don't often have spheres lying around, but boat hulls may have the same relationship they're like deformed spheres, right?
Assuming the boats are similarly shaped, the paint needed to coat one 50 foot yacht could instead paint a 40 and footer. In terms of energy,. With the energy used to accelerate one bullet to mph, we could accelerate two others to and mph. You can use any set of numbers that make a right triangle. For example, enter a total amount 50 and one subportion 30 , and the remainder will appear below:. Suppose you want to see if a large pizza 16 inches is bigger than two mediums 12 inches.
Plug in 16 for C, and 12 for A. It looks like the large pizza can be split into a inch and
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