Shortest Triangle Side

Date: 08/10/2002 at 14:53:46
From: Amanda
Subject: SAT math question - geometry related

A triangle has a base of length 13, and the other two sides are equal 
in length. If the lengths of the sides are integers, what is the 
shortest possible length of a side?

The answer key says the answer is 7. I know that the triangle would be 
isosceles and I tried working it out as a 45 45 90 and by drawing an 
imaginary line that is perpendicular to the base and using the 
Pythagorean theorem, but neither way came out to 7.

Help! Thank you.

Date: 08/10/2002 at 16:54:33
From: Doctor Paul
Subject: Re: SAT math question - geometry related

You have no basis to assume that this triangle will be a 45-45-90 
right triangle.  Doesn't it seem conceivable that the angle opposite 
the side of length 13 could be greater than 90 degrees?

This problem actually has nothing to do with the Pythagorean theorem.  
The concept is much simpler than that. And this is what makes it a 
hard problem. I think so much of what students see in Geometry-related 
questions on the SAT has to do with applications of the Pythagorean 
theorem that they've been conditioned to immediately think of the 
Pythagorean theorem when they see a triangle on the SAT.

Admittedly, this was my first thought as well. So evidently, we've 
been conditioned the same way.  :-)

But it's not the right way to approach the problem. It didn't take me 
long to realize that there was no reason to assume that this triangle 
contained a right angle. So there must be another way to approach this 
problem.

I reread the problem and then I saw immmediately what they were 
getting at. The fact that you gave me the answer definitely helped.

The problem says:

If the lengths of the sides are integers, what is the shortest 
possible length of a side?

Well, the first thing that occurred to me is that we know that the 
triangle could always be equilateral. So the triangle could have sides 
of length 13, 13, and 13. Since the SAT is a multiple-choice test, 
this immediately allows us to eliminate any answers greater than 13.  
You didn't tell me what the possible choices were, but I wouldn't be 
surprised if at least one of them was greater than 13. So even if you 
knew nothing about what I'm going to write below, you'd be able to 
eliminate some incorrect answers and you could guess from there.

Now we need to decide whether something smaller than 13 will do. The 
answer is of course in the affirmative. And seeing this requires that 
we become familiar with the so-called Triangle Inequality Theorem.

The Triangle Inequality Theorem says:

The sum of the lengths of any two sides of a triangle must be greater 
than the length of the third side. 

Perhaps the words are a bit confusing, but the concept is really quite 
obvious - if the third side had length 100 and the first two sides had 
lengths 10 and 15, I think you'd quickly come to the conclusion that 
these three lengths could not form a triangle. No matter how you 
tried, you'd never be able to "close" the third side because the first 
two sides just aren't long enough.

That's what they're getting at here.

The sum of the two congruent sides must be greater than 13. Since we 
know that the side lengths must be integers, that means that the sides 
must have lengths at least seven since 7 + 7 = 14 which, is greater 
than 13. Clearly, sides of length six (or shorter than six) will not 
do:

6 + 6 = 12 which is less than 13.

Thus the answer is 7.

I hope this helps. Please write back if you'd like to talk about this 
some more.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 08/11/2002 at 14:44:12
From: Amanda
Subject: Thank you (SAT math question - geometry related)

Doctor Paul -

Thank you for your help! I knew it was something easier than it 
seemed. That is not one of the obvious theorems that show up on the 
SAT, so that's why it had me stumped.

Thanks!