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Pythagoras – A Right Angle Review

CHAPTER 5

PYTHAGORAS – A RIGHT ANGLE REVIEW
(PART 1)

Pythagoras – A Right Angle Review will be similar to a story, even though the goal of this book is a geometrical pathway through spherical geometry, an assemblage of “discovery” data and its presentation for whatever of its purposes to future generations (only with a compass), it nevertheless appears that this pathway has certain concomitant destinations that are temporal, “ancient”, that cannot and should not be bypassed. It is necessary to pause and “have a talk” with persons who lived a long time ago before us because they knew much more even though for various reasons they failed to impart this knowledge to us. Hence, we have to talk with individuals and our contemporary generation should approach them with due respect because they are gifted with the baton of knowledge – the spirit of exploring – digging for us, for future generations – a fountainhead in the flinty rock of our existence to quench our thirst for fresh water, invigorate and be able to move further on to our beginning or ending, all the same. Therefore it sometimes hurts when we forget them, and even more when we don’t continue their work with the same goal in mind – for the coming generations and all the more when we realize that today there are those who call their efforts into question, and such persons are not just anyone, but persons who are in are acknowledged celebrities in our present-day world, who teach children and are rewarded for this, thereby producing a picture of this world as a world of perfection or finality, as if everything was created now and from now on, whereas everything that preceded was “primitive” or deserves to be “forgotten”. In earlier times people would sometimes ask me where is my “flock”? Then I would often tell them that we are “spread out” through various time periods and that one day all of us will “get together”. Today I am beginning to comprehend this more and more, even though this is not the place of our “gathering” but rather a meeting with a respected friend from ancient times – with Pythagoras –an encounter for a “chat” just to “remind” the forthcoming generation of the man who dedicated his life to the science of geometry and who left us his works. And from this “meeting” I hope to draw something new which he, for whatever reasons, did not leave behind.

That’s why I started out by saying that this will be similar to a story. Readers will understand and judge whether it is or isn’t important on basis of the first part of this “meeting” and surely detect my “partial” detour from the road of geometry only with a compass and unmarked straightedge, because of my use of measures so as to make things clearer for the contemporary concept of geometry. I will show the basic example of the right angle, the correct way of drawing it, wherefore and why, all based on the knowledge of someone led by a spirit or thirst for knowledge whom I approach with due respect for his effort and works – based on Pythagoras.

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How to draw a right triangle correctly.

Besides our use of the “angelic cubit” – 5.25 cm (1:10) we will have to apply the shortened variant of drawing (incomplete circles for a clearer presentation), but enough to comprehend with the basic right triangle of equal sides (the side as radius) and all this for a specific reason   – the height of the right triangle of equal catheti (legs) – in other words a straight line of specific length – the cathetus (leg).  

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Length – the cathetus (5.25 cm) as radius of the basic circle.

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The basic circle at the point where the straight line intersects it is divided by circles of same radius in order to acquire the other leg and right  triangle by using the intersections of the divisional circles.

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Thus we get a right angle (90°) and a cathetus (5.25 cm and 5.25 cm)

* * *

By connecting the peaks we get the hypotenuse (usually the divisional circles would also be divided up with divisional circles of the same radius, but we will do so only partially in order to get a parallel with the perpendicular leg).

* * *

And all this is to get an element, with respect to the height of the right triangle, because every triangle has a height that is vertical to the base line – in this case – the hypotenuse, but let us first calculate the length of the hypotenuse.

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Therefore, let’s call Pythagoras: Cathetus 5.25 squared = 27.5625 x 2 = 55.125 = sqrt{55,125} (second power of sum of squared cathetus) = 7.4246212. This means we have a triangle whose legs are the radius, and the hypotenuse is 1/4 square (read rectangle) of the circle of the cathetus.  

* * *

Why did we draw another line of symmetry? To get intersections, the symmetry axis of the hypotenuse – hence, the height of the leg of an isosceles right triangle.

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So, if the catheti are the sides of a right triangle then the height falls to the base line – the hypotenuse and with an isosceles triangle divides it into 2 parts and is thereby hipotenezu 1/2 of the hypotenuse height of such a triangle, in our case: hypotenuse = 7.4246212 ÷ 2 = 3.7123106 = H (Height).

* * *

If that is so, then the square root of the square height x 2 would give us the cathetus. Let’s check: square H = 3.7123106 squared  = 13.781249 x 2 = sqrt{27,562498} = 5.25 (true).

* * *

The straight line of the height intersects the arc of the cathetus circle. What’s that? What is the radius? Enter it into the compass from its peak – and divide the arc of the leg’s circle – data octagon (2 x 4-sided polygon).

* * *

And now that we know, with the help of Pythagoras, the size of the hypotenuse, we also know the magnitude and sides of the tetragon (hypotenuse). We can compute the squaring of the cathetus of the right triangle. The two squared hypotenuse = 55.125 = squared right triangle of circle. Divided by 4 = 13.781249 or squared H (Height) = squared cathetus of right triangle.

* * *

There is a series of other data but we find the relationship of the tetragon of the circle (its squaring) and squaring of the circle interesting.

* * *

So, here is the shortened variant:  2 x 5.25 – 10.5 x 10.5 = 110.25 ÷ 14 = 7.875 x 11 = 86.625 divided by 55.125 (squared tetragon) = 1.5714285 or in other words 11/7 and this means square of the circle divided by 11 (eleven) times 7 (seven) = 55.125 (quadrature of the square of the circle radius of the cathetus divided by 4 = 13.78125 (squared height of leg of right triangle = sqrt{13,78125}= height of  right isosceles right triangle x 2 =  its hypotenuse.

* * * *

Here we will stop with the data and summarize what we “found out” in this “talk” between me and Pythagoras.

The essential element is to correctly draw the right triangle (whether of equal or unequal catheti (legs).

  • In case of equal catheti the hypotenuse is equal to 2 H (two heights).
  • Each triangle has a height and it descends from the peak angle of the triangle vertically onto the baseline.
  • A series of data can be read from correct drawing of right triangles: computationally by means of the Pythagorean code and algebraically by means of a series of data as in this case
  • Squaring of the rectangle and its relationship to squaring of the circle, or vice versa, squaring of the circle by means of the hypotenuse of its construction 2H x 2H of right triangle of equal catheti is divided by 7 times 11 = squaring of the circle.

The next step would be a right triangle of unequal catheti, accompanying construction, algebraic and geometric relations.

The work of Pythagoras has a great share in all of this.

You may be sure that I will leave the last page geometrically unfinished (drawing of squaring the circle by radius of cathetus of the right triangle) since that should be the beginning of Part 2 of the right triangle (of unequal catheti). This does not contravene modern geometry but simply supplements it, makes corrections if necessary, or as the entity from above would say – New wine is good but old wine is better.

End of Part One of the Right Angle Review.

HR- RIJEKA
NOVEMBER, 2011.
Tomo Periša

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One Response to “Pythagoras – A Right Angle Review”

  1. Rod says:

    Oh, the power of those right angles of Pythagoras:
    http://www.aitnaru.org/images/Pi_Corral.pdf

    Geometers easily comprehend that this new concept of Pi simply complements one ratio (Pi) with another (ASR) and both ratios include the same mysterious and stimulating essence of irrationality!

    Such is the nature of squared circles.

    How not to square the circle?
    Believe that it is impossible.

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