Awesome Little Known Fact About Triangles: GRE and GMAT Tip

I have a secret about triangles that has been passed down to me from generation to generation since the ancients. Well, actually, I learnt it like 3 years ago. But still. It is a rule that I used in both my real GRE and GMAT tests.

I offer private GRE / GMAT tutoring online at a fixed rate of $140/hr. Please get in touch via the email below, or through my tutoring website: https://www.gretutorlondon.com/

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Each side of a triangle must be:
Less than the Sum of the Other Two Sides
And
Greater than the Difference Between the Other Two Sides

Before you rush off, close the video and tell all your friends, I advise you to see how this rule works in practice with a couple examples, and at the end I will show you how this rule can even be applied to shapes with four sides or more. Isn’t this so exciting?

First off, what does this rule really mean. Well, take a triangle with two sides of length 5 and 8. This doesn’t have to be a right-angled triangle, or an isosceles triangle, the rule applies to any triangle in the world.

If two of the sides are of length 5 and 8, the third side of the triangle must be less than 5+8, which is 13. And, the third side must also be greater than the difference between 5 and, which is 8 – 5 = 3. In other words, the third side of the triangle must be between 3 (less than) Third Side (less than) 13.

The reason this rule is true is because if the third side were 13 or more then the other two sides wouldn’t be able to stretch to reach that length, and you would have an incomplete triangle.


Likewise, if the third side were too short, 3 or less, then it couldn’t team up with the other short side to stretch and meet the longest side, again leaving us with an incomplete triangle.

It might seem like a simple rule, but very few students have heard of it, and if you know this rule confidently, you will have a real edge on the test.

Let me test you out.
If you have a triangle with two sides of lengths 11 and 15, which of the following could be the length of the third side of that triangle?
I. 4
II. 10
III. 24

The answer is 10 and 24 could be the third side but not 4. The sum of 11 and 15 is 26 and the difference between 11 and 15 is 4. So, the range of acceptable values for the third side is between 4 (less than) Third Side (less than) 26. Notice the less than sign, not less than or equal. 4 and 26 are not allowed.

Otherwise the triangle would be a perfectly flat line.
10 and 24 are within this range, but not 4.

OK, next test. If we have a triangle with side lengths of 7 and 14, which of the following could be the perimeter of that triangle?
I. 22
II. 33
III. 44
The answer is 33 only. If the two sides are 7 and 14, then the third side can be any number between 7 and 21. 7 is the difference, 21 is the sum. If the third side is between 7 and 21, the perimeter is between 7 + 14 + 7 and 7 + 14 + 21, or between 28 and 42. Only 33 is within that range.

Finally, let me test this concept in a different way. I am going to show you five sets of numbers representing the sides of a potential triangle. Give yourself 30-40 seconds to work out which of these five sets could not form a triangle.
6, 3, 4
7, 8, 13
8, 8, 8
6, 9, 16
8, 12, 4

The answer was the last two sets could not form a triangle: 6, 9, 16 and 8, 12, 4. 6 and 9 do not add to get 16 and 8 and 4 add to get exactly 12, which is not allowed.

OK, I think I have now convinced you that it is an amazing trick. Time to show you how it can extend to shapes that have four or more sides.

This time, there is no lower limit on the last side of the shape, but the upper limit is still the sum of all the other sides.

For example, with a pentagon, which has five sides, if the existing sides are of length 4, 5, 1 and 8, then the fifth side must be less than 18 (4 + 5 + 1 + 8 = 18). For a similar reason that the other sides wouldn’t be able to stretch and meet up with the final side if it was overly long.

There you have it! An awesome, maybe-ancient, maybe-not-ancient, trick about triangles. And trust me, this is tested on the GRE and on the GMAT.



I offer private GRE / GMAT tutoring online at a fixed rate of $140/hr. Please get in touch via the email below, or through my tutoring website: https://www.gretutorlondon.com/

Enquiries: [email protected] If you’re now ready to get into your dream MBA program, my former student, Angel, is offering virtual private Admissions coaching. She was admitted into Harvard Business School, The Wharton School of Business, and Columbia Business School (accepted into every school she interviewed for) after graduating from UCLA with a degree in Communication. She also got 3 perfect scores on the GRE’s Analytical Writing Assessment (99th percentile).

Here is her link - https://linktr.ee/angel_accel - and you can reach her at [email protected] with the subject line, “PHILIP SENT ME.”