Answer:
The possible lengths are:
2,3,4 and 5, 6, 7;
2,4, 5 and 3, 6, 7
2,6,7 and 3,4,5
Step-by-step explanation:
Because of the triangle inequality, we know that for any given triangle, the sum of two of its sides is always greater than the other side. And this is true for any two sides we pick.
Now we have segments of 2,3,4, 5, 6 and 7 inches long. So now we have to pair this segments into two groups of three in which the sum of two elements of each group is greater than the third element of said group.
There are 20 ways of choosing 3 elements out of a group of 6 elements, however, you should notice that the choosing of 2,3,4 is the same as the choosing of 5,6,7 (since by choosing any three elements for our first triangle the elements for our second one are determined and vice versa) Thus, we only have 10 different options to choose.
Option 1: 2, 3, 4 and 5, 6, 7
2 + 3 > 4
2 + 4 > 6
3 + 4 > 2
and
5 + 6 > 7
5 + 7 > 6
6 + 7 > 5
Thus, this is a possible side length of her two triangles
Option 2: 2,3,5 and 4, 6, 7
2 + 3 = 5
Thus, this is not a possible side length because it doesn't fit the inequality.
Option 3: 2,3, 6 and 4,5,7
2 + 3 < 6
Thus this is not an option.
Option 4: 2,3, 7 and 4,5 6
2 + 3 < 7
Thus, this is not an option either
Option 5: 2,4, 5 and 3, 6, 7
2 + 4 > 5
2 + 5 > 4
4 + 5 > 2
and
3 + 6 > 7
3 + 7 > 10
6 + 7 > 3
Thus, this is an option.
Option 6: 2,4, 6 and 3, 5, 7
2 + 4 = 6 therefore this is not an option.
Option 7: 2, 4, 7 and 3, 5, 6
2 + 4 < 7 thus this is not an option.
Option 8: 2, 5, 6 and 3, 4, 7
3 + 4 = 7, thus this is not an option
Option 9: 2, 5, 7 and 3, 4, 6
2 + 5 = 7, thus this is not an option.
Option 10: 2, 6, 7 and 3, 4, 5
2 + 6 > 7
2 + 7 > 6
7 + 6 > 2
and
3 + 4 > 5
3 + 5 > 4
4 + 5 > 3
Thus this is an option.
Therefore, our possible lengths are: 2,3,4 and 5, 6, 7; 2,4, 5 and 3,6,7; 2,6,7 and 3,4,5
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