 # Extra Hard SAT Math Question – Triangle Side Lengths

At the end of every SAT Math section, the test makers try to come up with an extremely difficult problem that will leave even the cleverest students scratching their heads. The really evil part, though, is that even these problems can be solved in under a minute without a calculator – if you know what to do. This means that once you “figure out the trick,” these difficult problems become easy. So, while those test makers are busy cackling with sadistic glee, let’s see if we can’t beat them at their own game.

Consider the following problem:

A triangle with one side 3 and another side 7 has perimeter P. What are the least and greatest possible integer values of P?

A) 5 and 9

B) 5 and 15

C) 9 and 15

D) 9 and 19

E) 15 and 19

Well, how on earth are we supposed to figure out the perimeter if we only know two sides? This is one where if you know how to do it, it’s a cinch, but if you don’t, it’s impossible. What you need to know for this problem is the Triangle Inequality:

“The triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.”

–Wikipedia

This principle is best understood visually: If the two shorter sides of the triangle equaled the longest side, then they would just lie flat on top of it and you’d have a line, not a triangle. Thus, the sum of the shorter sides must be less than the length of the longest side.

Returning to the problem at hand, we can now figure out the maximum and minimum lengths of the unknown side. If we want to find the maximum length of the unknown side, then we should let it be the longest side, which would mean that:

3 + 7 > longest side

10 > longest side

Thus, the longest side must be less than 10 units long. Remember, the problem asked for integer values, so that means the maximum length of the unknown side would be 9, since 9 is the first integer less than 10. To find the smallest that the unknown side could possibly be, we should assume that the longest side is the side with length 7. Let x represent the unknown side:

x + 3 > 7

x > 4

Thus, the shortest the unknown side could possibly be would be 5, since 5 is the first integer greater than 4. So the smallest the side can be is 5 and the largest it can be is 9. Choice A is the answer, right?

Wrong! Remember, the question asked for the minimum and maximum perimeters, not for the minimum and maximum lengths of the unknown side. Note that the minimum and maximum lengths of the unknown side were in one of the answer choices! To be extra tricky and evil, the test makers often put intermediate figures into wrong answer choices, hoping that you’ll forget what you were supposed to find in the first place and that you’ll stop working as soon as you get some numbers that match an answer choice. Don’t fall for this common trap! Always make sure you found what the problem was actually asking for before you mark your answer.

Finding the minimum and maximum perimeters is itself quite easy. The minimum perimeter would be:

3 + 5 + 7 = 15

and the maximum perimeter would be:

3 + 7 + 9 = 19

Thus, the correct choice is E, 15 and 19. If you know what to do, it takes only about 30 seconds to solve this problem. So you see, with practice, even the hardest problems on the SAT become easy. Check back here each week for more extra hard problems and the tricks you need to solve them! Also, remember that you can find out all the tricks from experts like me with a Test Masters course or private tutoring. Until  then, keep up the good work and happy studying!

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1. Anna Haliman says:

Actually, this is simple. You know no side can have a negative length. So, add 3 and 7 to get 10. Thus, the minimum value has to be 10 or more. There is only one to which this applies.

1. Bill says:

Hi Anna! Could you more clearly explain your strategy here? I’m interested in seeing what your approach was!

1. John says:

It’s simple: once you add the lengths of the first two sides and get the sum of 10, you know that the perimeter must be greater than 10, since the third side can’t be zero. The only answer with the perimeter greater than 10 is E, which is 15.

2. DCW says:

It merely follows the triangle inequality theorem: two sides when added must be greater than the third side, but when the smaller is subtracted from the other, the third side must be larger than the difference.

So, in the problem, the missing side could be from 5 to 9.
1. 7+3=10, the value that the third side must be less than
2. 7-3=4, the value that the third side must be greater than
3. x, the third side is as follows: 4<x<10
Therefore, x is between 5 and 9, with its smallest and largest sides being 5 and 9 respectively.
The perimeter's smallest value is 15=7+3+5 and its largest value is 19=7+3+9

It takes much less time to do when you are solving it just mentally, or mentally and on paper!