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### Possible Areas of a Triangle

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Date: 12/27/2001 at 18:25:01
From: Sahil Bhargava
Subject: Possible Areas of A Triangle (SAT)

Dear Dr. Math,

I encountered the following problem while studying for the SAT:

One side of a triangle has length 6 and a second side has length 7.
Which of the following could be the area of this triangle?
I. 13
II. 21
III. 24

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I, II, and III

The answer, according to the back of the book, is D, and while it is
clear to me that II is possible because the height could be 6 and
base 7 in a right triangle, I wasn't able to get further into this
problem.

Thanks for the help.
Sahil Bhargava
```

```
Date: 12/27/2001 at 19:04:59
From: Doctor Achilles
Subject: Re: Possible Areas of A Triangle (SAT)

Hi Sahil,

Thanks for writing to Dr. Math.

you're missing. Here's a hint:

Start with a right triangle that has 6 and 7 as the two legs. Next,
draw one more right triangle with 6 as a side and 7 as the hypotenuse.
What is the area of this?  (A right triangle with 7 as a side and 6 as
the hypotenuse is impossible.)

Next, draw a non-right triangle with 7 as one side and 6 as the base.

What will the height of the triangle be? Or a better question:
Can the height of the triangle be 7?  Can it be less than 7?
Can it be more than 7?

Finally, draw a non-right triangle with 6 as one side and 7 as the
base.

What will the height be? Can the height be more than 6? Can the height
be arbitarily small (could you make a triangle with a height of, say,
0.0001 if you wanted)?

So what is the minimum area you can make given two sides of a
triangle? What is the maximum area?

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

```
Date: 12/28/2001 at 15:25:45
From: Sahil Bhargava
Subject: Possible Areas of A Triangle (SAT)

Thanks a lot, Dr. Math. I understand how to progress now.
```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Triangles and Other Polygons

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