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### Triangle Proof

```
Date: Sat, 18 Feb 1995 22:01:29 -0800 (PST)
From: Ralph Sierra

I'm a student in high school and I need help on this one problem.
I'm in geometry and stumped. I'm thinking that maybe to figure it
out, it needs to be said that if 2 sides of a triangle are not
congruent, then the angles opposite them are not congruent, and the
larger angle is opposite the longer side. But I'm not sure how to
say it in a proof. I have an idea but am getting a little confused.
If you could help me, I would appreciate it.

A
/|\
/ | \
/  |  \
/   |   \
/    |    \
/     |     \
/      |      \
/       |       \
/        |        \
/_________|_________\
B           D           C

B, D and C are points of the line L such that B-D-C and BD<DC.
If segment AD is perpendicular to L, prove that AB<AC.

Thank you,
- Michelle
```

```
Date: Sun, 19 Feb 1995 02:16:52 -0500 (EST)
From: Dr. Ken

This is a job for the law of Sines.  As you may know, if A, B,
and C are the angles of a triangle and a, b, and c are the sides
opposite them, then

a         b       c
------ = ------ = ------
Sin[A]   Sin[B]   Sin[C].

So in your case, if what you need to show is that AB<AC,
a good way to do that would be to show that angle C is smaller
than angle B. Then you'd be able to say that

AB          AC
-------- = --------
Sin[C]     Sin[B],

If you buy that, then all you have to do is show that angle C
is smaller than angle B. I hope this train of thought gets you
somewhere, and write back if you're still stumped.

-Ken "Dr." Math
```

```
Date: Sun, 19 Feb 1995 08:50:45 -0800 (PST)
From: Ralph Sierra

In our class we have not studied the law of sines yet, so
I'm not following your directions.  I'm sure you're correct,
but can you please explain it a little simpler?  Thanks

- Michelle
```

```
Date: Sun, 19 Feb 1995 12:57:34 -0500 (EST)
From: Dr. Ken

Hello there.

Not allowed to use the law of sines, huh?  Well, perhaps we can
do this with the Pythagorean theorem.  Oh, wow, in fact it's
easiest this way.

We know that BD<CD.  Using the Pythagorean theorem
(BD^2 + AD^2 = AB^2 and CD^2 + AD^2 = AC^2), what can you conclude
about the lengths of AC and AB?  I think this is an easier way
than the way I was trying to explain before.

-Ken "Dr." Math
```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons

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