Surprising Uses of the Pythagorean Theorem


#143

hi Kalid hope you have time:

so if i see c as a vector that can be split into components (a,b) i can just as well see the area defined by c as a vector that can be split into components of the areas defined by a and b.
so when we split the right triangle into two self-similar triangles, we really split the area into the components of its growth in the two orthogonal dimensions…

what do you think of this? i found this really beautiful but i just wonder how to think of it.

then what we see as the area is really the magnitude of the area vector just as the hypotenuse is just the magnitude of the vector c. and what is magnitude? in fact it is just projection in a different coordinate system…
somehow this is really whirling up quite a bit of my understanding of direction. pretty interesting…


#144

Hi Khalid, I noticed that you applied the relationship for the well known 5x5 = 4x4 + 3x3 Pythagorean to circular areas. Did you know that 6x6x6 = 5x5x5 + 4x4x4 + 3x3x3 would similarly apply to spherical volumes?


#145

Hi Nicolay, great question. It’s figure-out-able after some work but not “obvious” (in that you should see it immediately). Thanks for working through it, it’ll help people!


#146

Okay, I just realised the answer to my question - which in retrospect I should’ve inferred from the title of the post.

For anyone wondering:

d = hypotenuse
s = the length of the sides

So we have:
d^2 = s^2 + s^2
d^2 = 2s^2
1/2 * d^2 = s^2
d/sqrt(2) = s

… I guess it was kinda obvious. :stuck_out_tongue:


#147

A three-points geometric model of Pi?

Consider this remarkable Pythagorean triangle that defines both a circle and its square. The center of the circle is located at the point where the perpendicular to the hypotenuse connects with the triangle’s long side; circle’s diameter = 4:

Long side = Pi = 3.14159265358979323846264…
Hypotenuse = (square root of Pi) x 2
= 3.54490770181103205459633…
= side length of circle’s square.

The related circle-squaring inscribed scalene triangle is inherent in this Pythagorean geometry. The scalene’s short side has length equal to one side of a square inscribed in the circle; the longest side forms a 45-degree angle with the hypotenuse of the right triangle.


#148

Hei Kalid!

Another great article, man! One question: How does one intuitively infer that “A regular side is d/sqrt(2)” where d is the diagonal?


#149

Wow, that blew my mind! I’d always thought the Pythagorean Theorem was kind of dull, but you gave it a whole new twist. :slight_smile: And the whole Area=factor*(line segment)^2 was also pretty amazing. I’d never thought of it that way before. Keep up the good work.


#150

Thanks Yana, glad you enjoyed it!


#151

i dont know what to say its helped me a lot


#152

I think the reason the kinetic energy theorem works is this: The force of an object is mass times acceleration, you square the velocity and multiply by 0.5 because acceleration is measured in squares and velocity is the dirivive of acceleration so to balance out the equation you divide by 2.


#153

I find difficult 2 find the uses of the Pythagoras theorems extension…can sum 1 plzz help me find it… Plzz…


#154

How can you write 5 =4+3 ???


#155

Advantages of using pythagorean theorem are not discussed in the notes above.


#156

hi Kalid, I’m a new math teacher. Please recommend a good project about Pythagorean theorem for my students to do.


#157

Thank you so much.


#158

not useful at all


#159

Awesome website, a dose of mathematical sanity, this is the real thing - mathematical meaning before mathematical manipulation (though sometimes we seem to need to learn the manipulation first). (By the way Karan (No. 122), Kalid is not saying “5 = 4 + 3”, but “the area of a circle radius 5 = area of a circle radius 4 + area of a circle radius 3”.)


#160

Two Mango’s of 300 gram and 400 gram equal to One mango of 500 gram? assume the seeds are proportional!!


#161

Brilliant article Kalid, it gives wings to ur imagination!!! i would suggest adding some practical uses of this theorem as i had came to this page looking for some daily uses of the theorem but ended up getting amazed by it… :smiley:


#162

http://TrianCal.esy.es - Open in Google Chrome. (Triangles online calculator developed by Jesus S.)
YouTube: https://youtu.be/V2IV7lY52mA

I propose this free online calculator triangles without advertising to help students with geometry, does not perform the duties, because their calculations formulas are not displayed. It is designed in a didactic way to check and view the realized duties.

TrianCal is online calculator triangles that works with any combination of values including sides, heights, angles, the area or perimeter of any triangle, calculating it with the minimum possible value (typically three).

Other functions:

  • Draw the triangle (s) with GeoGebra.
  • Set the range of values in each element.
  • The type of angle.
  • The type of triangle by its angles and sides.
  • Selection of language (English or Spanish).
  • Select the angle type [degrees (°), radians, degrees, minutes and seconds (° ’ ") or degrees and minutes (° ')].
  • Number of decimal places shown in the results (0-15).
  • You can use the arrow keys and the Tab key to navigate through the settings.
  • Drop-down menu to select the values comfortably.
  • Create a link (URL) to the current triangle.
  • An icon mail to communicate with the author.

NOTE: You must use the Google Chrome browser to display correctly TrianCal.

Examples of possible combinations:

  • The area, perimeter and other data (side, height or angle), if the outside equilateral triangle would not need the third data.
  • 2 angles and other data (if the value of the other data is not put aside the value of “a” at the time of drawing the triangle is 10).
  • One side, one high and one angle.
  • 3 heights.
  • 3 sides.
  • 2 heights and perimeter.
  • Any other combination of values.