Guide to the Triangle Inequality Theorem

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The Triangle Inequality Theorem is one of those things that only seems scary because it has an intimidating name, like Occam’s Razor, Arpeggios, or Nnamdi Asomugha. But the Triangle Inequality Theorem is actually very easy, and if they gave it a simpler name- say, the Triangle Side Thingy- everybody would get these questions right on the ACT, SAT, GRE, and GMAT.

The picture above illustrates the rule nicely. You can’t make a triangle when one side is larger than the other two sides put together. The two smaller sides couldn’t possibly connect to each other and still reach the edges of the longer side. Therefore, the Triangle Side Thingy is as follows: Any side of a triangle must be smaller than the sum of the other two sides.

Another way of looking at it, from the point of view of one of the smaller sides, is that it must be at least large enough to make up the gap between the other two sides. In other words, a side of a triangle must be larger than the difference between the other two sides. Putting it all together, what you need to remember for the SAT, ACT, GRE, and GMAT is Any side of a triangle must be smaller than the sum of the other two sides and larger than the difference between the other two sides.

Here’s an example of how the Triangle Side Thingy might be tested on the SAT, ACT, GRE, or GMAT:

1.) In a triangle with sides of lengths 3 and 6, which of the following is a possible value of the perimeter of the triangle?

A.) 10

B.) 12

C.) 16

D.) 18

E.) 20

The Triangle Side Thingy tells us that whatever the third side of the triangle is, it must be larger than 3 (the difference of the other two sides) and smaller than 9 (the sum of the other two sides). A perimeter is the sum of all three sides. Here, only 16 is possible because the third side could be 7. Choices A, B, D, and E are impossible because they would require the third side to be 1, 3, 9, or 11 respectively. The answer is choice C.

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