Decimal Fractional Parts of Measured Dimensions on Hypotenuses of Right Right Triangles

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Decimal Fractional Parts of Measured Dimensions on Hypotenuses of Right Right Triangles

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 fractional, or decimal part of the positive real number 3. The decimal fractional part, .75 = the fraction 3/4.

For sources, see links below;

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Dimensions with fractional, or decimal parts, invariably occur on the two segments of the hypotenuse of a right triangle, caused by the altitude to the hypotenuse.
In the link below, see segments "p" and "q".

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NOTE: Segments "p" and "q" are used in this post to help visualize graphically the association with actual dimensions calculated here for the identical type of segments on the hypotenuse.

For example; SA1, SB1 & SC1 corresponds to the greater segment "q", and SA2, SB2 & SC2 corresponds to the lesser segment "p".

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

Refer to Post 1 for description, and construction of triangles A and B. Triangle C is a known baseline right triangle, from which A & B are derived.

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

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

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

2. Segments SA1, SA2 on Hyp of A
83.20754717 Greater Seg SA1.
6.79245283 Lesser Seg SA2.
- - - - - - - -.- - - - -
90.00000000 = (2×45)*
See Items 10 and11 for segment calculations.

* 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.

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

4. Altitude to Hyp in C=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.Triangle A has 1/2 the lesser acute angle in C. Triangle B has 1/2 the greater acute angle in C. See NOTE above.

8. Use formula below for Greater Seg SB1 on Hyp of B.

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.*

*em<>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 (2x45) of C. See NOTE above.

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

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

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

SC2 = a^2 / h
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. 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
                                            
42.79245283 = SB1
13.20754717 = SB2
- - - - - - - - - - - -
56.00000000 = 2x28

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

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

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

15D. The greater segment SA1 on the hypotenuse of A plus the greater segment SB1 on the hypotenuse of B is the Perimeter of C.

83.20754717 SA1
42.79245283 SB1
- - - - - - - - - - - -
= 28 + 45 + 53 = Perimeter of C.

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 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. They turned out to be identical to those on the hypotenuse segments of the baseline triangle C.

FOCUS:
The focus of this post is to call  attention to the unconventional
and intriguing way of calculating areas of right triangles created in the same manner as "A" and "B". It is unconventional in the sense that it applies only to those types of  right triangles.

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.

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.

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

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