# Find perimeter of traiangle when you know two sides

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.

• If the other side is longer than the first two put together, there's no way for the first two sides to reach from one end of the third to the other end. Similarly, if the other side is shorter than the difference of the first two, you have the same problem with the longer of the first two sides. – user2357112 Apr 21 '14 at 3:32
• $12+18+x=\text{perimeter}$ but $18-12<x<12+18$ – Piquito Jan 2 '16 at 22:01

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.

The third side must be between $18-12 = 6$ and $18+12 = 30$.

• But none of those choices are between 6 and 30. Unless you add them this way: $18 + 12 + 6 = 36$ and $18 + 12 + 30 = 60$. Those two values are in between 36 and 60. – Alex Apr 21 '14 at 3:23
• The third side must be between those numbers, therefore the perimeter must be between $30 + 6 = 36$ and $30 + 30 = 60$ – Juan Sebastian Lozano Apr 21 '14 at 3:25
• How did you approve this suggested edit? That was one hell of a misclick. – Daniel Fischer Jan 2 '16 at 22:15

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