# PWN the SAT Q&A

Anonymous

Triangle ABC has a perimeter of 10, and the lengths of its sides are all integers. If a is the length of side BC, what is the difference between the largest and the smallest possible values of a? (A) 1 (B) 2 (C) 3 (D) 4 (E) 7

This question is about the triangle inequality theorem, which states that no triangle side can be longer than the sum of the other two sides.

To solve this without getting too intense with equations and inequalities, take a step back and think logically about how big a could possibly be. Remembering that it can’t be bigger than the sum of the other two sides, that all the sides have integer lengths, and that the sides must add up to 10, there aren’t actually that many possibilities.

It couldn’t be 5, for example. If the largest side was 5, then the other two sides would have to add up to 5 for the perimeter to be 10, but that wouldn’t work with the triangle inequality theorem! Say you had sides of 5, 3, and 2. You’d really just have overlapping segments of length 5, not a triangle. Take a minute to think about this paragraph. It’s important.

So the longest a side can be is 4. And in that case, its other sides must add up to 6.

What are all the ways a triangle can be made with a longest side of 4, integer lengths of the other two sides, and a perimeter of 10?

4, 4, 2
4, 3, 3

And…that’s it. So we’ve figured out the shortest a side could be, too: 2.

The longest a side could be is 4, and the shortest it could be is 2. Difference: 2.