Here's the GRE problem: The length of one side of a triangle is 12. The length of another side is 18. Which of the following could be the perimeter of the triangle?
[ ] 30
[ ] 36
[X] 44
[X] 48
[ ] 60
Why are C and D correct? I'm reading that for the perimeter of a triangle, the length of the 3rd side must be between the positive difference and the sum of the other two sides. I'm lost as to what that means? Can someone help? Thanks.
Call the three sides $a=12$, $b=18$, and $c$. The "triangle inequality" says a straight line is the shortest route between two points, i.e. to get from one vertex of the triangle to another, just follow the side that connects them and you've got the shortest route; if you take the other two sides, it's longer. In other words: \begin{align} a+b & \ge c \\ b+c & \ge a \\ c+a& \ge b \end{align}
So \begin{align} 12+18 & \ge c \\ 18+c & \ge 12 \\ c+12& \ge 18 \end{align}
This tells us $c\ge 18-12 = 6$ and $c\le 12+18 = 30$.
The perimeter is $a+b+c=12+18+c$. This is therefore $\ge36$ but $\le60$.
It appears that exactly $36$ and exactly $18$ are ruled out because "triangle" is take to mean nondegenerate triangle, i.e. the three vertices are not on a straight line.