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### 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.

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!
```
Associated Topics:
High School Triangles and Other Polygons

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