The Triangle Rule

I consider using the triangle rule to be an advanced math strategy for standardized tests. It generally comes up on level 4 and 5 problems on the SAT, ACT and GRE, and the only reason so many students get these problems wrong is because they have never been taught the rule in school.

The triangle rule states that that the length of the third side of a triangle is between the sum and difference of the lengths of the other two sides.

So, for example, if we knew that a triangle had two sides of lengths 5 and 7, respectively, then we could find the possible lengths of the third side by using the triangle rule. Simply note that the sum is

7 + 5 = 12

and the difference is

7 – 5 = 2.

Therefore, the the length of third side lies between 2 and 12.

Let’s get our heads around this with another short example:

Example 1

If a triangle has sides of lengths 2, 5, and x, then we have that

5 – 2 < x < 5 + 2.

That is,

3 < x < 7.

The triangle rule is a very easy concept to understand. Again, the only reason why these are considered tough problems is because the rule is not emphasized in school, and sometimes not taught at all.

So let’s really get this rule into practice by solving some problems that might actually appear on a standardized test.

Example 2

The lengths of the sides of a triangle are x, 8, and 15, where x is the shortest side. If the triangle is not isosceles, what is a possible value of x?

Solution: The triangle rule tells us that

15 – 8 < x < 15 + 8.

That is,

7 < x < 23.

Since x is the shortest side, x < 8. So we must choose a number between 7 and 8. If this were a grid in problem (such as on the SAT or GRE), we could grid in 7.1 or any other decimal or improper fraction between 7 and 8. But inputting 7 or 8 as the answer would be incorrect as these integers are not between 7 and 8!

Okay, one more. This one is a multiple choice question.

Example 3

If x is an integer greater than 5, how many different triangles are there with sides of length 3, 5 and x?

A) One
B) Two
C) Three
D) Four
E) Five

Solution: The triangle rule tells us that

5 – 3 < x < 5 + 3.

That is,

2 < x < 8.

Since x is an integer greater than 5, x can be 6 or 7. So there are two possibilities, choice B.

My students find these problems very straight forward. I hope you do too. But it can be easy to forget the rule you need to use if you do not practice enough with it. If you want more practice, then please check out the  Get 800 collection of test prep books. These have more examples of problems that need to use this rule for efficient and correct answering.

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