Tangled Triangles

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.

Then I went on to work out the combinations for 60 o as the angle and then for 80 o. This left me with the first nine combinations.

The next task was to work out all the combinations using two angles. I started by using 40 and 60 which gave three combinations, then 60 and 80 and then finally 80 and 40. Now I had explored all the possible combinations of two angles and was left with nine more combinations, three for each. 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. In order to find this the first task is to work out the missing sides and the missing angles to work out the area and perimeter. This can be done by labelling angles A, B, and C and then labelling the opposite sides a, b and c respectively. Then the Sine or Cosine rule can be used depending on which values are available and which need to be found.

Below are the first eight of the triangles with the working out shown. 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.)