**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

**segments on the hypotenuse of three dissimilar right triangles.**

*greater*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.

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

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

identicalto those on the hypotenuse segments of the baseline triangle C.Not only that, the recurring decimal fractional components

switchedsegments on the hypotenuses of A and B.That is, when the

lessersegment SA2 (6.79245283) onAcontained thegreaterdecimal fraction, then onB, thegreatersegment SB1 (42.79245283) contained the samegreaterdecimal 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

Ais equal to the greater segmentSB1, on the hypotenuse ofB, mutiplied by the difference between the hypotenuse ofCand the short leg ofC.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.

:PROPOSITION2. The area of

Bis equal to the greater segmentSA1, on the hypotenuse ofA, mutiplied by the difference between the hypotenuse ofCand the long leg ofC.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

shortleg of C multiplied by the qreater segmentSC1, on hypotenuse ofC, and divided bylongleg of C, equals the area ofB.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.

PROPOSITION4. The square of the

longleg of C multiplied by the lesser segmentSC2, on hypotenuseC, and divided by theshortleg of C, equals the area ofA.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

Bis equal to the (sum of the two legs ofCminus the hypotenuse ofC), multiplied by the (sum of the altitude to hypotenuse and the greater segmentSB1), 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 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 trianglesA&B, (derived fromC).FOCUS:This post addresses certain unique mathematical curiosities observed in those triangles, as stated below;

1. The sum of the greater Seg's

SA1andSB1on hypotenuses of trianglesAandB, respectively, equals the perimeter of triangleC.SA1 = 83.20754717.

SB1 = 42.79245283.

83.20754717 + 42.79245283 =

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

2.

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