To solve this question, we should know the Triangle Inequality Theorem.
The triangle inequality theorem says something like: The sum of any two sides of a triangle must be greater than than the length of the third side
Making that into a formula, considering the side lengths are a, b, and c.
a + b > c
a + c > b
b+ c > a
So:
Lets test answer choice A.
The sides will then be: 34, 12, and 21
34 + 12 > 21 -> 46 > 12 True
12 + 21 > 34 -> 33 > 34 False
A is not the correct answer choice.
Lets try answer choice B.
The sides for this would be: 34, 12, and 30
34 + 12 > 30 -> 46 > 30 True
12 + 30 > 34 -> 42 > 34 True
30 + 12 > 34 -> 42 > 34 True
So B is correct.
Lets try C to make sure.
The side lengths would be: 34, 12, and 46
34 + 12 > 46 -> 46 > 46 False
So C is wrong.
Lets just make sure D is also wrong, and then we will know for sure if B is the answer :)
Side lengths of answer choice D: 34, 12, and 50
34 + 12 > 50 -> 46 > 50 False
So B is indeed our answer. :)
Another way to solve it is to:
We are given side lengths of 34 and 12.
That means the other side has to be greater than the different of 34 and 12 and less than the sum of 34 and 12.
Lets say that c is the third side, then:
34 -12 < c < 34 + 12
22 < c < 46
And by looking at this range, and the answer choices we know that B) 30 is the answer because the others are at the border and since it isn't greater than or equal to/less than or equal to, we can not include it.Or either the answer choices are out of the range.
So the final answer is [tex]\boxed{\bf{B)~30~cm}}[/tex]
