Triangle inequality theorem

The Triangle Inequality Theorem can be used to find the minimum and maximum possible lengths of a missing side of any triangle. Sometimes this is referred to as the Third Side Rule or the 3rd Side Rule.

Consider this triangle below, with two sides of $4$ and $9$.

We haven’t gone over the rule yet, but take a moment to consider what lengths the missing side could be.

We don’t know anything about the angles of this triangle, but we’re still able to determine a range for the missing side. How can that be? It’s the Triangle Inequality Theorem that allows us to find this range of values quickly.

Let’s come back to our example triangle.

Checking the rules, the missing side $x$ must be:

1. Greater than $9-4=5$

   This minimum side length is easier to imagine since it’s drawn like the figure. The triangle with the smallest third side would have a tiny, almost-zero angle between the $4$ and $9$ sides, almost like a straight line with the $4$ and $9$ sides overlapping. If the missing side were any smaller than $5$ we would be left with a gap, and if the side was precisely $5$, we wouldn’t have a triangle at all - it would be a straight line.

2. Less than $9+4=13$

   Remember, figures on the GRE aren’t drawn to scale. The maximum side length occurs when the angle between the $4$ and $9$ sides is as large as possible, with the $4$ side to the left and the $9$ side on the right end-to-end, almost in a straight line but with an angle slightly smaller than $180^{\circ }$. If the missing side were greater than $13$ it would be too long and wouldn’t connect - we’d have an extra line segment hanging off the triangle. And if it were precisely $13$, we’d have a straight line, not a triangle.

So $x$ must be greater than $5$, and less than $13$, i.e. in the range $5<x<13$.

A GRE question may directly ask you for the missing side of a triangle, but it might also ask something more complicated, like this example question below.

Which of the following could be the perimeter of the triangle shown in the figure?

Select all that apply.
A. 9
B. 10
C. 12
D. 14
E. 17

Can you apply the Triangle Inequality Theorem to figure it out?

First, we can solve for the range of the missing side using the Triangle Inequality Theorem. The difference between the two sides is $5-3=2$, and the sum is $5+3=8$. This means that the missing side could be anything within the range of $2<x<8$.

Using this knowledge, we can figure out the range of the perimeter. The perimeter is simply the sum of the sides. The two sides we’re given sum up to $5+3=8$. Using the bounds of $x$, we can find the bounds of the perimeter:

• Since the third side $x$ must be greater than $2$, the perimeter must be greater than $5+3+2=10$.
• Since the third side $x$ must be less than $8$, the perimeter must be less than $5+3+8=16$.

Combining these bounds, the range for the perimeter is $10<p<16$.

Keep in mind that the range is exclusive and does not include $10$ or $16$. If we used $2$ or $8$ as the third side in the triangle, we wouldn’t end up with a triangle - we would have a straight line instead.