The Geometry Rule That's More Important Than You Think

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This tip for improving your GMAT score was provided by David Newland at Veritas Prep.

There is one very simple rule that students seem to have trouble remembering and even more difficulty applying: the Third-Side Rule.

The essence of the rule is that to have an actual triangle, no one side of the triangle can be longer than the other two sides put together. Imagine a triangle with one side length 10 and the other two sides have lengths of 3 and 5. There is simply no way that the two shorter sides could connect. You would not have a triangle.

The actual rule is stated in terms of the third side (which is why it is called the third-side rule): “The third side of a triangle must be smaller than the sum of the other two sides and must be greater than the difference between the other two sides.”

Use the third side rule to solve the following question quickly and efficiently from the GMATPrep software: “If two sides of a triangle have lengths of 2 and 5, which of the following could be the perimeter of the triangle?

I. 9
II. 15
III. 19

A) None
B) I only
C) II only
D) II and III
E) I, II, and III”

Apply the first part of the rule to determine the maximum value that the perimeter could equal. Given that the third side—in this case, the unknown side—must be smaller than the sum of the two known sides, and since that sum is 7 (2 + 5), the third side of the triangle must be smaller than 7. Therefore the perimeter must be less than 14.

The second part of the rule establishes the minimum for the unknown third side. Since the third side must be greater than the difference between the two known sides, and since that difference is 3 (5 – 2), the third side must be bigger than 3. Therefore the perimeter must be more than 10.

None of the available options is between 10 and 14, so none could be the third side of this triangle. Therefore the correct answer is A: “none.”

A surprising number of GMAT questions rely on the third-side rule. Some questions, such as the one we just solved, clearly ask for the third side, while others are cleverly disguised. An excellent example of a question that is made so much easier by the third-side rule is this question, which also appears courtesy the GMATPrep software: “The perimeter of a certain right isosceles triangle is 16 + 16 √2. What is the length of the hypotenuse?

A) 8
B) 16
C) 4 √2
D) 8 √2
E) 16 √2”

Most people can set up the algebra on this problem but often do not know where to go from there. If you look at this one logically from the start, you can see where the third-side rule applies.

An isosceles right triangle has two equal sides and a hypotenuse that is √2 larger. Use x to represent each of the equal sides, and you get the equation, “2x + x √2 = 16 + 16 √2.” Many test takers look at this equation and think that the hypotenuse must have the square root; most people choose 16 for the sides and 16 √2 for the hypotenuse. Unfortunately, choice E is the trap answer on this question.
Using the third-side rule could help you quickly find the correct answer. Begin by recognizing that the √2 can be attached to either the sides or the hypotenuse but not both. Either the sides are an integer, such as 10, and the hypotenuse is 10 √2, or the sides have the square root, such as 2 √2, and the hypotenuse is the integer 4.

Now apply the rule: The sum of the two sides must be greater than the length of the third side – the hypotenuse. So the 16 √2 (which you can estimate to be approximately 23) must be the sum of the two sides and 16 must be the length of the hypotenuse. Otherwise, that ~23 would be too large for the two smaller sides, since they’d add up to 16. Therefore, the correct answer is B.

The third-side rule is one of the simplest rules to memorize, but it can also be one of the most important.

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