Possible Areas of a Triangle

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've actually almost got the answer already. There's just one step 
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?

I hope this helps. If you have other questions about this or you're 
still stuck, please write back.

- 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.