File:FS ER dia.png
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Summary
[edit]DescriptionFS ER dia.png |
English: Largest rectangular triangle within a equilateral triangle
Deutsch: Größtes rechtwinkliges Dreieck in einem gleichseitigen Rechteck |
Date | |
Source | Own work |
Author | Hans G. Oberlack |
Licensing
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Task
[edit]The equilateral triangle as base element.
Inscribed is the largest right triangle.
General case
[edit]Segments in the general case
[edit]0) The side length of the equilateral base triangle is:
1) The side length of the inscribed right triangle is: , see calculation (2)
Perimeters in the general case
[edit]0) Perimeter of equilateral base triangle:
1) Perimeter of inscribed right triangle:
Areas in the general case
[edit]0) Area of the equilateral base triangle: , see calculation (1)
1) Area of the inscribed right triangle:
Centroids in the general case
[edit]0) By definition the centroid point of a base shape is
1) The centroid point of the inscribed right triangle relative to the centroid of the base shape is: see calculation (3)
Normalised case
[edit]In the normalised case the area of the base shape is set to 1.
So
Segments in the normalised case
[edit]0) Side length of the triangle
1) The side length of the inscribed triangle is:
Perimeters in the normalised case
[edit]0) Perimeter of base triangle:
1) Perimeter of inscribed triangle:
S) Sum of perimeters:
Areas in the normalised case
[edit]0) Area of the base triangle is by definition
1) Area of the inscribed triangle:
Centroids in the normalised case
[edit]0) By definition the centroid point of a base shape is
1) The centroid point of the inscribed triangle relative to the centroid of the base shape is:
Distances of centroids
[edit]The distance between the centroid of the base triangle and the centroid of the inscribed triangle is:
Sum of distances:
Identifying number
[edit]Apart of the base element there is one other shape allocated. Therefore the integer part of the identifying number is 1.
The decimal part of the identifying number is the decimal part of the sum of the perimeters and the distances of the centroids in the normalised case.
So the identifying number is:
Calculations
[edit]Known elements
[edit](0) Given is the side length of the equilateral triangle:
(1)
(2)
(3)
Calculation 1
[edit]The height is calculated:
,applying the Pythagorean theorem on the rectangular triangle
, applying equation (2)
, applying equation (1)
, rearranging
, rearranging
, rearranging
The area of triangle is height multiplied by the half of the side length:
, applying equation (2)
Calculation 2
[edit]The side length is calculated:
,applying the Pythagorean theorem on the rectangular, isosceles triangle
,applying equation (3)
,applying equation (1)
, rearranging
, drawing the root
, rearranging
With this result the height of triangle can be calculated:
, applying the Pythagorean theorem on the rectangular triangle
, applying equation (2)
, applying equation (3)
, applying the first part of calculation (2)
, rearranging
, rearranging
, extracting the root
Calculation 3
[edit]
, since is in the centre of the figure
, definition of centroid in an equilateral triangle
, see calculation (1) for height
, multiplying
, definition of centroid in an isosceles triangle
, see second part of calculation (2)
, rearranging
, rearranging
, definition of centroid of the base shape
, rearranging
File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 22:13, 16 September 2023 | 1,229 × 1,260 (50 KB) | Hans G. Oberlack (talk | contribs) | Uploaded own work with UploadWizard |
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