Triangle inequality
The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the third side.
Let's analyze each group of side lengths:
• 7,6,10
Note that:
7+6 > 10
7+10 > 6
6+10 > 7
Since all the conditions are met, this group can create a triangle
• Now for 4,2,7
Testing each sum:
4+2 > 7
This inequality is false since 6 is not greater than 7
Thus, this group cannot create a triangle
• Now for 3,4,5
Testing each sum:
3 + 4 > 5
3 + 5 > 4
4 + 5 > 3
Since all the conditions are met, this group can create a triangle
• Finally, for 3,3,3
Testing each sum:
3 + 3 > 3
This happens three times. They form in fact an equilateral triangle
Thus, the only group that cannot create a triangle is 4,2,7