Cool ways to use the Pythagorean Theorem

In summary, the Pythagorean Theorem is a mathematical principle that states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. It has numerous applications, including finding the "normal lens" for cameras, generating pi, and proving time dilation in special relativity. There are over 370 known proofs for this theorem, and finding new proofs is considered a cool thing to do. It can also be used in multiple dimensions and even infinite dimensions with certain limitations. Additionally, it is used in the vector addition of real and reactive power in electrical engineering. However, it is also possible to disprove this theorem, as well as other well-known theories and concepts.
  • #1
Hatesmondays
17
9
What are some cool things that people can do with the Pythagorean Theorem?
latex.php?latex=a%5E2+%2B+b%5E2+%3D+c%5E2+&bg=ffffff&fg=2b2b2b&s=3.png
 
Physics news on Phys.org
  • #2
Prove it.
 
  • #3
Prove what?
 
  • #4
The theorem.
 
  • #6
The last time I used it was to find the "normal lens" for my cameras.

Here's the wiki article:

http://en.wikipedia.org/wiki/Normal_lens

Basically, the "normal" lens for any given camera is that lens which is middle of the road, being neither wide angle nor telephoto. What focal length that is for any given camera is dictated by the film or sensor size. It is that focal length which is equal to the length of the diagonal of the format or sensor. Knowing the sensor width and height, it is a simple matter to use Pythagorean theorem to find the diagonal.
 
  • #7
Hatesmondays said:
Pythagorean Theorem
The theorem states that:

"The square on the hypotenuse of a right triangle is equal to the sum of the squares on the two legs"
Here are 109 other proofs: http://www.cut-the-knot.org/pythagoras/
There are at least 370 of them, according to wiki. Including one by U.S. President James Garfield.
Hence, finding new proofs are a cool thing to do. Q.E.D.
 
  • #8
Enigman said:
There are at least 370 of them, according to wiki. Including one by U.S. President James Garfield.
Hence, finding new proofs are a cool thing to do. Q.E.D.
Would it be possible, using certain foundational results in mathematics or logic that I am shamefully unaware of, to establish an upper bound on the number of different proofs? (For this one of course first has to specify when two proofs are considered "different".)
 
  • Like
Likes Silicon Waffle
  • #9
I used it (iteratively) to generate π once.
 
  • #10
Hatesmondays said:
What are some cool things that people can do with the Pythagorean Theorem?
latex.php?latex=a%5E2+%2B+b%5E2+%3D+c%5E2+&bg=ffffff&fg=2b2b2b&s=3.png
The sum of the squares of the standard deviations of two independent random variable is equal to the square of the standard deviation of their sum.
 
  • Like
Likes collinsmark
  • #11
Hatesmondays said:
What are some cool things that people can do with the Pythagorean Theorem?
latex.php?latex=a%5E2+%2B+b%5E2+%3D+c%5E2+&bg=ffffff&fg=2b2b2b&s=3.png

I like that works with any number of dimensions. a^2 + b^2 + c^2 + ... + y^2 = z^2

It is even used in infinite dimensions, with the proviso that the sum has to be finite.
 
  • #12
Hornbein said:
I like that works with any number of dimensions. a^2 + b^2 + c^2 + ... + y^2 = z^2

It is even used in infinite dimensions, with the proviso that the sum has to be finite.
I like that one.
 
  • #13
A Mathematical Fable

 
  • Like
Likes Matejxx1
  • #14
I've been working with Power Factor a lot lately in my EE work. The Pythagorean Theorem comes up in the vector addition of Real and Reactive Power:

http://www.rbgrant.co.uk/wp-content/uploads/2012/11/Power-Factor-Correction-2.jpg [Broken]
http://www.rbgrant.co.uk/wp-content/uploads/2012/11/Power-Factor-Correction-2.jpg [Broken]

http://www.rapidtables.com/electric/electric_power.htm
Real / reactive / apparent powers relation
The real power P and reactive power Q give together the apparent power S:

P^2 + Q^2 = S^2

P is the real power in watts [W]

Q is the reactive power in volt-ampere-reactive [VAR]

S is the apparent power in Volt-amper [VA]
Power factor definition
The power factor is equal to the real or true power P in watts (W) divided by the apparent power |S| in volt-ampere (VA):

PF = P(W)/ |S(VA)|

PF - power factor.

P - real power in watts (W).

|S| - apparent power - the magnitude of the complex power in volt·amps (VA).

A power factor of 1.0 is ideal, and devices with input Power Factors < 1.0 are starting to have to use "Power Factor Correction" to bring up their low numbers... :smile:
 
Last edited by a moderator:
  • #15
Tiling a floor - Measure one side (A) at 10 feet and mark both ends, at one of the ends of A make a perpendicular arc 10 feet long. On the other end of A, measure 14.14 feet and scribe an arc to intersect with the first arc.
 
  • #16
berkeman said:
I've been working with Power Factor a lot lately in my EE work. The Pythagorean Theorem comes up in the vector addition of Real and Reactive Power:

http://www.rbgrant.co.uk/wp-content/uploads/2012/11/Power-Factor-Correction-2.jpg [Broken]
http://www.rbgrant.co.uk/wp-content/uploads/2012/11/Power-Factor-Correction-2.jpg [Broken]
Ooh ooh, I remember this from physics! We used an inductor to improve efficiency! (decrease the non-doing reactive power)
 
Last edited by a moderator:
  • Like
Likes berkeman
  • #17
You can effectively proof the time dilation formula from Special Relativity using little more than Pythagoras. That might be considered cool.
 
  • #18
Borek said:
Prove it.

It's even cooler to disprove it!
 
  • Like
Likes Borek
  • #19
That is right! I'm am going to try to disprove it. On another site cuase this one doesn't allow new ideas.
 
  • #20
Hatesmondays said:
That is right! I'm am going to try to disprove it. On another site cuase this one doesn't allow new ideas.

If you enjoy to disprove stuff that is true, I can give you some other suggestions:
- Disprove the theory of evolution (this is a popular one, googling will get you a lot of references!)
- Disprove germ theory
- Disprove gravity
- Disprove that Earth is round
- Disprove that 1+1=2
- Disprove you exist
 
  • #21
micromass said:
- Disprove that 1+1=2
Let [itex]1\in\mathbb{Z}_2[/itex], then [itex]1+1 = 0[/itex] :D :D
 
  • #22
nuuskur said:
Let [itex]1\in\mathbb{Z}_2[/itex], then [itex]1+1 = 0[/itex] :D :D

Not a disproof since 0=2 in ##\mathbb{Z}_2##.
 
  • Like
Likes PWiz, collinsmark and nuuskur
  • #23
Unfortunately true, but worth a shot.
 
  • #24
micromass said:
If you enjoy to disprove stuff that is true, I can give you some other suggestions:
...
It is very easy to disprove the pythagorean theorem! First, bring a ball ...
Not going to finish the proof to avoid spoilers :biggrin:

Edit:
Although I suppose that an assumption of the theorem is that you are not allowed to bring a ball
 
  • Like
Likes nuuskur
  • #25
micromass said:
- Disprove you exist

I exist?
 
  • #26
Boom there.
 
  • #27
Hatesmondays said:
I exist?
Hatesmondays said:
Boom there.
The fact that' you're even able to even ask that question proves [to you] that you exist. (Proves it to yourself, that is. It doesn't necessarily prove it to anybody else; that's a significantly more difficult problem. But if you're capable of asking yourself about your own existence, it proves to yourself that you exist. https://en.wikipedia.org/wiki/Cogito_ergo_sum.)
 
  • #29
Hatesmondays said:
I exist?

I didn't ask to prove you exist, I asked to disprove you exist.
 
  • #30
micromass said:
I didn't ask to prove you exist, I asked to disprove you exist.

I don't think, therefore I am not.
 
  • #31
Well...
 

1. How can the Pythagorean Theorem be used to find the distance between two points?

The Pythagorean Theorem can be used to find the distance between two points on a coordinate plane by using the formula c = √(a^2 + b^2), where c represents the distance between the two points and a and b represent the lengths of the sides of a right triangle formed by the two points.

2. Can the Pythagorean Theorem be used to solve real-life problems?

Yes, the Pythagorean Theorem can be used to solve real-life problems such as finding the length of a ladder needed to reach a certain height on a wall or the distance a person needs to travel to get from one point to another. It is a useful tool in fields such as architecture, engineering, and navigation.

3. How can the Pythagorean Theorem be used to find the length of a diagonal in a rectangle?

The Pythagorean Theorem can be used to find the length of a diagonal in a rectangle by using the formula c = √(a^2 + b^2), where c represents the length of the diagonal and a and b represent the lengths of the sides of the rectangle. This is because the diagonal of a rectangle forms the hypotenuse of a right triangle with the sides being the length and width of the rectangle.

4. Is the Pythagorean Theorem only applicable to right triangles?

Yes, the Pythagorean Theorem can only be applied to right triangles, which are triangles with one angle measuring 90 degrees. This is because the theorem is based on the relationship between the sides of a right triangle and the length of its hypotenuse.

5. Can the Pythagorean Theorem be used in other areas of mathematics?

Yes, the Pythagorean Theorem has many applications in other areas of mathematics such as trigonometry, geometry, and algebra. It is also a fundamental concept in the study of the Pythagorean triples, which are sets of three positive integers that satisfy the theorem.

Similar threads

Replies
2
Views
876
  • General Math
Replies
8
Views
2K
  • Precalculus Mathematics Homework Help
Replies
13
Views
910
Replies
53
Views
8K
  • Quantum Interpretations and Foundations
Replies
2
Views
2K
Replies
72
Views
4K
  • Precalculus Mathematics Homework Help
Replies
17
Views
1K
  • Calculus
Replies
12
Views
2K
Replies
5
Views
680
Back
Top