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Question 282344: which ways can I make a triangle with 25 centimeters of perimeter?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! The answer is that, if you allow the length of a side to be any real number, then the number of ways is infinite.
One example will show you why:
Let's assume the length of the largest side of the triangle is 12 cm.
This means that the sum of the lengths of the two other sides has to be equal to 13 cm.
Let's assume length of one of the smaller sides is 8 cm.
This means the length of the other of the smaller sides is 5 cm.
The sum of the lengths of all 3 sides of this triangle is 12 + 8 + 5 = 25 cm.
Now let's take another possibility:
The length of one of the smaller sides is 7 cm and the length of the other of the smaller side is 6 cm.
The sum of the lengths of all 3 sides of this triangle is 12 + 7 + 6 = 25 cm.
We have at least two ways we can make a triangle with a perimeter of 25 cm.
In between these two triangles there is an infinite number of ways.
One of the smaller sides can vary between 7 and 8 cm in length while the other of the smaller sides can vary between 5 and 6 cm in length.
Some examples would be triangle with the following lengths:
12, 8, 5
12, 7.9, 5.1
12, 7.8, 5.2
12, 7.7, 5.3
12, 7.6, 5.4
12, 7.5, 5.5
etc.
Between a triangle having sides of 12, 8, 5 and a triangle having sides of 12, 7, 6, there are an infinite number of possible points in between.
You can divide 1 cm into 2, 4, 6, 8, 16, 32, 64, 128, 512, ...... x number of pieces, where there is no limit to how large x can grow.
Bottom line is there are an infinite number of ways if you assume that the length of each side can be any real number.
If you restrict the length of each side to be something like an integer, then there are countable number of ways in which you can get a triangle to have a perimeter of 25 cm.
Assume the same 12 cm for the largest side of the triangle.
The sum of the lengths of the other 2 sides of the triangle would have to be 13 cm in order for the perimeter to be equal to 25 cm.
This can happen in the following ways:
12,12,1
12,11,2
12,10,3
12,9,4
12,8,5
12,7,6
Assume the largest side is 11 and you get:
11,11,3
11,10,4
11,9,5
11,8,6
11,7,7
Assume the largest side is 10 and you get:
10,10,5
10,9,6
10,8,7
Assume the largest side is 9 and you get:
9,9,7
9,8,8
Assume the largest side is 8 and you get:
8,8,9 which can't be, because this would make 8 NOT he largest side of the triangle, so we stop at the largest side of the triangle having a minimum length of 9.
As you can see, now the number of ways are countable.
This is not the case when the length of the side of a triangle can be any real number.
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