Decimal Fractional Parts of Dim's on Hypotenuses of Dissimilar Right Triangles

 
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Decimal Fractional Parts of Dim's on Hypotenuses of Dissimilar Right Triangles

DECIMAL FRACTIONS:
The decimal, fractional part of a positive real number is the excess beyond that number's integer part; e.g., for the number 3.75, the numbers to the right of the decimal point make up the decimal, fractional part of the positive real number 3. The decimal fractional part, .75 equals the fraction 3/4.

For sources, see links below;

https://en.wikipedia.org/wiki/Decimal?wprov=sfla1.

https://en.wikipedia.org/wiki/Fractional_part?wprov=sfla1.

EXPLANATION:
Dimensions with decimal, fractional parts invariably occur on the two segments of the hypotenuse of a right triangle, caused by the altitude to the hypotenuse. The product of these segments equals the square of the altitude to the hypotenuse. See Items 15E thru 15H.

In the link below, see altitude to hypotenuse h and segments "p" and "q".

https://en.wikipedia.org/wiki/Geometric_mean_theorem?wprov=sfla1.

NOTE: Segments "p" and "q" are used in this post to help visualize graphically their association with actual dimensions calculated here for the lesser and greater segments on the hypotenuse of three dissimilar right triangles.

For example, hypotenuse segments; SA1, SB1 & SC1 on dissimilar right triangles A, B & C, correspond to the greater segment "q", and segments SA2, SB2 & SC2 correspond to the lesser segment "p".

ABOUT THIS POST; This post deals with the incredibly obscure, and recurring identical decimal fractions, of actual calculated dimensions of the two segments on the hypotenuses of 3 dissimilar right triangles, A, B & C, caused by the altitude to the hypotenuse, as defined in Geometry, Post 1.

Hypotenuse segments are considered 
irrational numbers. In fact all square roots of natural numbers, other than of perfect squares, are irrational.

They are also all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers, like the number 3.75 at the top of page.

Irrational numbers may also be dealt with via non-terminating continued fractions. They can be represented in precisely one way as an infinite continued fraction. See following links:

https://en.wikipedia.org/wiki/Line_segment?wprov=sfla1.

https://en.wikipedia.org/wiki/Continued_fraction?wprov=sfla1.

Triangle C can be any right triangle, and is known as the baseline triangle, from which A & B are derived, as stated in Geometry, Post 1, "Three Dissimilar Right Triangles".

Refer to the above post for description, and construction of triangles A and B.

PROPOSITION: The decimal fractional part of the dimension on the greater line segment "SA1" on hypotenuse of A is equal to the decimal fractional part of the dimension on the lesser line segment "SB2" of the hypotenuse of B, and the greater line segment "SC1" on the hypotenuse of C.

PROPOSITION (continued)
The decimal, fractional part of the dimension on the lesser line segment "SA2" on hypotenuse of A is equal to the decimal fractional part of the dimension on the greater line segment "SB1" of the hypotenuse of B, and the lesser line segment "SC2" on the hypotenuse of C. See Item 15.

1. For baseline triangle C, Pythagorean triangle 28, 45, 53 is chosen. See Items 12 thru 14 for calculations of segments SC1, SC2.

2. Segments SA1, SA2 on Hyp of A
83.20754717 Greater Seg SA1.
06.79245283 Lesser Seg SA2.
- - - - - - - -.- - - - -
90.00000000 = (2×45)*
See Items 10 and11 for segment calculations, accurate to 8 decimal places.

* Post 1, "Three Dissimilar Right Triangles" requires Hyp of A to be twice the long leg (45) of C. See NOTE above.

3. Segments SB1, SB2 on Hyp of B
42.79245283 Greater Seg SB1.
13.20754717 Lesser Seg SB2.
- - - -.- - - - -.-.- -
56.00000000 = (2×28)*
See Items 8 and 9 for segment calculations, accurate to 8 decimal places.

* Post 1, "Three Dissimilar Right Triangles" requires Hyp of B to be twice the short leg (28) of C. See NOTE above.

4. Let "d" = Altitude to Hyp in C. d = 28x45/53 = 23.77358491.

5. Altitude to Hyp in A and B is same as that in C*.

* Post 1, "Three Dissimilar Right Triangles" requires altitude to hypotenuse in A & B to be identical to that in C. See NOTE above.

6. Acute angles in A & B are 1/2 the acute angles in baseline triangle C.

7. For instance; Triangle A has 1/2 the lesser acute angle in C. Triangle B has 1/2 the greater acute angle in C.

8. Use formula below for greater Seg SB1 on Hyp of B; a = short leg of C; b = long leg of C; c = hypotenuse of C; d = altitude to hypotenuse of C.

SB1= b x d / (c - a)
SB1=45 x 23.77358491/(53 - 28) =
42.79245283. See Item 9 for Seg SB2. See Item 3.*

9. SB2 on Hyp of B = 56.00000000 - 42.79245283 = 13.20754717. See Item 3.*

* Post 1, "Three Dissimilar Right Triangles" requires Hyp of B to be twice the short leg (28) of C. See NOTE above.

10. Use formula below for greater Seg SA1 on Hyp of A. See Item 2.

SA1= a x d / (c - b).
SA1=28x23.77358491/ (53 - 45) = 83.20754717. See Item 11 for SA2. See Item 2.*

11. SA2, on Hyp of A = 90.0000000 - 83.20754717 = 6.79245283. See Item 2.*

* Post 1, "Three Dissimilar Right Triangles" requires Hyp of A to be twice the long leg (45) of C. See NOTE above.

12. For greater Seg SC1 on Hyp of C, use formula below; where b = long leg of C; c = hypotenuse. See Item 1.

SC1 = b^2 / c
SC1 = 45^2 / 53 = 38.20754717.

13. For lesser Seg SC2 on Hyp of C, use formula below, where a = short leg of C; c = hypotenuce. See Item 1.

SC2 = a^2 / c
SC2 = 28^2 / 53 = 14.79245283.

14. For sum of SC1, SC2. See below;

38.20754717 = SC1.
14.79245283 = SC2.
- - - - - - - - - - - -
53.00000000. See Item 1.

15. The following is a comprehensive review of the above, comparing segment dimensions and their decimal fractional parts. See Proposition at top.

83.20754717 = SA1
6.79245283 = SA2
- - - - - - - - - - - -
90.00000000 = 2x45 = Hyp of A.
                                            
42.79245283 = SB1
13.20754717 = SB2
- - - - - - - - - - - -
56.00000000 = 2x28 = Hyp of B.

38.20754717 SC1
14.79245283 SC2
- - - - - - - - - - - -
53.00000000 = Hyp of C.

15A. The decimal fraction .20754717 of Seg SA1 equals the decimal fraction of SB2 & SC1. See Proposition.

15B.The decimal fraction .79245283 of SA2 equals the decimal fraction of SB1 & SC2. See Proposition.

15C. The sum of the decimal fractions .79245283 and .20754717 equals 1.00000000.

15D. The square root of the products of the greater and lesser segments, SA1, SA2 and SB1, SB2, on each of the hypotenuses of triangles A & B, is equal to the square root of the product of the greater and lesser segments SC1, SC2 on the hypotenuse of triangle C.

15E.The above quantity is known as the Geometric Mean between the 2 segments on the hypotenuse of a right triangle. Also known as the altitude to the hypotenuse, as described in the following link;

https://en.wikipedia.org/wiki/Geometric_mean_theorem?wprov=sfla1.

15F.The altitude to the hypotenuse of any right triangle is equal the product of the 2 legs divided by the hypotenuse. For example; in baseline triangle C (28, 45, 53); Alt to Hyp = 28 x 45 / 53 =
23.77358491.

NOTE: The above quantity was used in Items 8 &10 to calculate the greater segments SA1 & SB1.

15G. The square of the altitude to the hypotenuse of any right triangle is equal to the product of the 2 segments on the hypotenuse. For triangles A, B & C, that product is 565.1833393.

15H. The square root of 565.1833393
is 23.77358491, the altitude to the hypotenuse on triangles A, B & C. See Items 4, 5 and 15F.

Tags: 

To the best of my knowledge,

Comment #2, created by BobDH, deleted 04/26/18.

In Post 1, a study

In Post 1, a study investigated the dimensions of segments on the hypotenuses of dissimilar right triangles A, B & C.

Triangle "C", a known baseline triangle (28, 45, 53), and two triangles A & B, (derived from C), were used in Post 1, to demonstrate the unique and improbable outcome of the study. 

Improbable, because of the uncanny recurring decimal fractional components of the segment dimensions on the hypotenuses of A & B, and the fact they were identical to those on the hypotenuse segments of the baseline triangle C.

Not only that, the recurring decimal fractional components switched segments on the hypotenuses of A and B.

That is, when the lesser segment SA2 (6.79245283) on A contained the greater decimal fraction, then on B, the greater segment SB1 (42.79245283) contained the same greater decimal fraction. See Item 15, Post 1.

FOCUS:
The focus of this post is to call  attention to the unconventional and bizarre way of calculating areas of right triangles created in the same manner as "A" and "B", as described
in Geometry, Post 1 of Three Dissimilar Right Triangles"

PROPOSITION:
1. The area of A is equal to the greater segment SB1, on the hypotenuse of B, mutiplied by the difference between the hypotenuse of C and the short leg of C.

For Example; C = 28, 45, 53 triangle.

Hyp of C = 53.
Long leg of C = 45.
Seg SB1= 42.79245283.

Area of A = 42.79245283 (53 - 28)
Area of A = 1,069.811321

Conventional Method:
Area of A equals the hypotenuse multiplied by the altitude to hypotenuse, divided by 2.

Hyp of A = 90.*
NOTE: Triangle A has same altitude to Hyp as C.*
Altitude to Hyp of C = 28 × 45 / 53 = 23.77358491.

* See Post 1.

Area of A = 90 x 23.77358491 / 2 =
1,069.811321.

PROPOSITION:
2. The area of B is equal to the greater segment SA1, on the hypotenuse of A, mutiplied by the difference between the hypotenuse of C and the long leg of C.

Hyp of C = 53.
Long leg of C = 45.
Seg SA1= 83.20754719.

Area of B = 83.20754719 (53 - 45).
Area of B = 665.6603775.

Conventional Method:
Area of B equals the hypotenuse multiplied by the altitude to hypotenuse, divided by 2.

Hyp of B = 56.*
NOTE: Triangle B has same altitude to Hyp as C.*
Altitude to Hyp of C = 28 × 45 / 53 = 23.77358491.

Area of B = 56 x 23.77358491 / 2 =
665.6603775.

* See Post 1 above.

NOTE: Area A : Area B :: 45 : 28.

PROPOSITION:
3. The square of the short leg of C multiplied by the qreater segment SC1, on hypotenuse of C, and divided by long leg of C, equals the area of B.

For example; C = 28, 45, 53 triangle.

SC1 = 38.20754717. See Post 1.

Area B = 28^2 x 38.20754717 / 45 = 665.6603775. See Proposition 2.

PROPOSITION
4. The square of the long leg of C multiplied by the lesser segment SC2, on hypotenuse C, and divided by the short leg of C, equals the area of A.

For example; C = 28, 45, 53.

SC2 = 14.79245283. See Post 1.

Area A = 45^2 x 14.79245283 / 28 = 1,069.811321. See Proposition 1.

PROPOSITION:
5. The area of B is equal to the (sum of the two legs of C minus the hypotenuse of C), multiplied by the (sum of the altitude to hypotenuse and the greater segment SB1), divided by two.

For example; C = 28, 45, 53.

Alt to Hyp = 28 × 45 / 53 = 23.77358491.

Seg SB1 = 42.79245283. See Proposition 2.

Area of B= (28 + 45 - 53)(23.77358491 + 42.79245283) / 2

Area of B = 665.6603774. See Proposition 2.

Unheard-of Curiosities of

Unheard-of Curiosities of Three Dissimilar Right
Triangles
.

EXPLANATION:
In Post 1, dimensions of segments on the hypotenuses of 3 dissimilar right triangles were investigated. Triangle "C", a known baseline triangle (28, 45, 53), and two right triangles A & B, (derived from C).

FOCUS:
This post addresses certain unique mathematical curiosities observed in those triangles, as stated below;

1. The sum of the greater Seg's SA1 and SB1 on hypotenuses of triangles A and B, respectively, equals the perimeter of triangle C.

SA1 = 83.20754717.
SB1 = 42.79245283.

83.20754717 + 42.79245283 =
126.00000000 = 28 + 45 + 53 = Perimeter of C.

2.

Unheard-of Curiosities of

Post 5, a duplicate of Post 4, was removed 06/02/18.

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