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Tangled Triangles

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Introduction

Tangled Triangles The first thing to do in solving this problem was to draw all the possible triangles which can be made using the given measurements of 40, 60 and 80. When I had exhausted all the possible combinations and eradicated ones which where the same I was left with 20 triangles. (see next two pages) I went about finding all the possible triangles by firstly starting off with one angle. The first angle I used was 40o and I worked out all the possible side combinations, which turned out to be 3 different combinations. But there are different looking combinations which are actually the same so I had to be careful when doing this (see below) These two triangles may look as if they are different combinations but if one of them was flipped over it would look exactly the same as the other. ...read more.

Middle

When I was doing this I had to be very careful not to repeat the same triangle because the same triangle can often look different the other way round. For example: These two triangles look quite different but if either is rotated then they will look the same. Now I had 18 triangles and there were just two more possibilities that I could see; a triangle with 40, 60 and 80 as all the sides and a triangle with these angles. This brought the total number to 20 triangles and there were no other combinations that were not just the same triangles flipped over or rotated. These 20 different triangles can be seen on the next 2 pages. The task states that the triangle with the biggest value of Area / Perimeter must be found. ...read more.

Conclusion

The dimensions, angle sizes, area, perimeter and Area / Perimeter for the remaining triangles are all shown on the table on page 7 with the exception of triangles 6, 9 and 19 which can't be worked out. Triangle 1 Triangle 2 Triangle 3 Triangle 4 Triangle 5 Triangle 6 Triangle 7 Triangle 8 Now that I have compiled a table of results I am able to state that the triangle that gives the biggest value of Area / Perimeter is triangle 12 which is illustrated below. Although this is the biggest value of Area / Perimeter for the 17 triangles I was able to obtain results, one of the three triangles for which nothing could be calculated could have a bigger value than this. The illustration below shows the missing angles and sides for this triangle. (All sides are in centimetres and all angles are in degrees.) ...read more.

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