From triangle inequalities, the sum of any two side lengths of a triangle must be greater than the length of the third side. Given three side lengths, add together the two smaller lengths. If the sum is greater than the third side, such a triangle is possible to form. However, if the sum is less than or equal to the length of the third side, it is not possible to form such a triangle. This is because the three sides would not be able to be connected to form a polygon.
Using what we know about triangle inequalities, that is, any two sides of a triangle must sum to more than the third side, we can determine just by looking at the lengths if a triangle is possible.
So here we have a triangle whose side lengths are 12, 13 and 21. The only thing you need to do is say, well the two shorter sides if I add those up will be more than the third side. Well, 12 plus 13 is 25, and 25 is greater than 21. So the answer to this one would be yes, you can make a triangle with sides 12, 13 and 21.
Let’s look at another one. If I look at 7, 8 and 15. If I add up the two shorter sides, 7 plus 8 I get 15. Is that inequality true, 15 is greater than 15? No, so the answer to this one would be no.
Now, let’s say I changed this just a little bit and made this 8.1. That would make this 15.1 in which case our answer would be yes, since 15.1 is greater than 15. All you have to do, add up your two shorter sides, if it is more than your third side, you’re good to go and you have a triangle.