Source: Official Guide Revised GRE 1st Ed. Part 6; Set 2; #13


If the lengths of two sides of

If the lengths of two sides of a triangle are 5 and 9, respectively, which of the following could be the length of the third side of the triangle? Indicate all such lengths. 3, 5, 8, 15

3 Explanations


Roshan Shrestha

The only important idea I haven't been able to grasp is what if we had the lengths 4 and 14 in the options. Are they the valid lengths of the third side, or the third side can only be in between the range 4 and 14 ? I mean can't we say the triangles with third side equal to either 4 or 14 as triangles with area 0 ?

Jan 20, 2017 • Comment

Sam Kinsman, Magoosh Tutor

Hi Roshan,

No, 4 and 14 wouldn't work. In order to have a triangle, the sum of the length of the two shortest sides must be GREATER THAN the length of the longest side.

If we tried to build a triangle with sides 5, 9, 14, we would not be able to do that, because 5 + 9 = 14. So we would just have two line segments on top of a third line segment. I can see why this would seem like it's a "triangle" with area 0 - but it would not have 3 distinct sides, which is necessary in order for us to have a triangle.

Jan 20, 2017 • Reply


Sean McGarry

I think that the 3rd side rule is a much easier way of going about this. In any triangle, 1 side has to be greater than the differences of the other 2 sides and less that the sum of the other 2 sides. Therefore, in this case, side 3 has to be greater than 4 and less than 14.

Aug 30, 2014 • Comment

Caiti Rothenberg

Just one clarification: technically, the side must be greater than the ABSOLUTE VALUE of the difference of the other two sides.

In this example, 5-9 = -4. Using common sense, one would likely realize it's not possible to have a negative value for length. BUT, the reason this lack of clarification could be dangerous is that if someone were taking this wording of the rule literally (without thinking critically about it), then there are a number of positive values greater than -4 that are not actually possible values for the third side of this triangle (such as 1, 2 or 3).

Jan 24, 2016 • Reply

Cydney Seigerman, Magoosh Tutor

Good point! Yes, all of the side lengths must be positive. With that in mind, when we apply the Triangle Inequality Theorem, we usually subtract the smaller side from the larger side (9-5 = 4). It is good to keep in mind, though, that if someone were to subtract the larger side from the smaller side, then the difference will be negative and that the third side must be larger than the absolute value of the difference.

Jan 26, 2016 • Reply


Gravatar Chris Lele, Magoosh Tutor

Oct 7, 2012 • Comment

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