Deriving the ratio of the hypotenuse is quite easy, we just apply Pythagorean theorem:
\(a^2+b^2=c^2 \)
\(1^2+1^2=c^2\)
\(\sqrt{2}^2-b^2=1^2\)
\(Replaced \ c \ with \ \sqrt(2)\)
\(-b^2=1^2-\sqrt{2}^2\)
\(b^2=\sqrt{2}^2-1^2 \)
\(b=\sqrt{2}-1\)
Because 1² equals 1… 1²+1² would just be 1 + 1 (2) equaling c².
\(c^2=2\)
\(c=\sqrt{2}\)
Now we want to derive the square unit,
Let a = 5
Knowing the length of the hypotenuse can be expressed as a*sqrt(2), we can say that the hypotenuse (c) equals 5*sqrt(2)=7.0710
Now, the length of side (b) as a ratio of the line, extended horizontally, is as follows…
\(⬛️ \)
\(5*\sqrt{2}-5 \ \ expressed \ as...\)
\(5*(\sqrt{2}-1)\)
Finding the area of the side b would give us a square unit to multiply in covering the area.
From the last article, ‘Area of a Square’,
\(b*(\sqrt{2}-1)^2 \)
We go to determining the square unit. With the help of the simplified square formula l, derived…
\(3-2\sqrt(2)\)
That equals 0.1715, this is our divisor.
Assuming we have a base and height equal to 1 we can use that square unit as a bit to process the area as follows…
\(\frac{1}{3-2*\sqrt(2)}= 5.8284 \)
That square unit can be applied to base width of 1,2,3,4,5,5.8284
For example,
Let b = 2
\(\frac{2}{3-2*\sqrt(2)}\)
Equaling (=) 11.65
In terms of the relationship in between the base and the hypotenuse, it's significant… We come to a mathematical proof algebraically, that we have 1/32nd.
Finally, a mathematical proof
\(\frac{1}{3-2*\sqrt(2)}= 5.8284 \)
5.8284²
Must equal
\( \frac{5.8284}{3-2*\sqrt{2}} \)
The result equals 33.97, which is 5.8284 squared!
Now, we know the area of a square by multiplying b base * (h) 5.8284 until we reach base equaling 5.8284.
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