File:FS RQ2 dia.png
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Contents
Summary
[edit]DescriptionFS RQ2 dia.png |
English: Second largest square inscribed in a right isosceles triangle
Deutsch: Zweitgrößtes Quadrat, das in ein rechtwinkliges gleichschenkliges Dreieck eingeschrieben ist |
Date | |
Source | Own work |
Author | Hans G. Oberlack |
Task
[edit]The second largest square inscribed in a right isosceles triangle of side length
General case
[edit]Segments in the general case
[edit]0) The side length of the base triangle is:
1) The side length of the inscribed square is: , see Calculation (1)
Perimeters in the general case
[edit]0) Perimeter of base triangle:
1) Perimeter of inscribed square:
Areas in the general case
[edit]0) Area of the base triangle:
1) Area of the inscribed square:
Centroids in the general case
[edit]0) By definition the centroid point of a base shape is
1) The centroid point of the inscribed square relative to the centroid of the base shape is:
, see calculation (2)
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 square is: ,
Perimeters in the normalised case
[edit]0) Perimeter of base triangle:
1) Perimeter of inscribed square:
S) Sum of perimeters:
Areas in the normalised case
[edit]0) Area of the base triangle is by definition
1) Area of the inscribed square:
Centroids in the normalised case
[edit]0) By definition the centroid point of a base shape is
1) The centroid point of the inscribed square 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 semicircle 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)
(4)
(5)
(6), since is parallel to and therefore is similar to
(7)
Calculation 1
[edit], since is similar to
, applying equation (4)
, applying equation (1)
, rearranging
applying equation (5)
, applying equation (6) and
Calculation 2
[edit]The centroid of the square relative to the centroid of the triangle is:
, using the formular for the centroids of triangles
, applying equation (6) and </math>|AD|^2+|AG|^2=2\cdot|AD|^2=a_1^2
, applying equation (4)
, applying equation (7)
, shortening
, applying calculation (1)
, shortening
Licensing
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File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 09:19, 2 October 2022 | 811 × 731 (18 KB) | Hans G. Oberlack (talk | contribs) | improved version uploaded | |
21:59, 28 September 2022 | 663 × 601 (14 KB) | Hans G. Oberlack (talk | contribs) | Uploaded own work with UploadWizard |
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