area of a triangle-

[xc630]

Hi I need some help with this problem.

The problem says suppsoe a trinagle has 2 sides of lengths a & b. if the angle bewteen these sides varies, what is the maximum and minimum possible areas that the triangle can attain.

I think I have to use 1/2 ab sin C but Im not sure. I would appreciate any help.-----

 

[pka]

The area, A=(1/2)(ab)sin(C), is correct.
Now 0<C<pi; so what value of C means that the sin(C) in maximum?
Note that the minimum is approaching zero.

 

[soroban]

Hello, xc630!

Suppsoe a trinagle has 2 sides of lengths \(\displaystyle a\) and \(\displaystyle b\).
If the angle bewteen these sides varies,
what is the maximum and minimum possible areas that the triangle can attain?

I think I have to use \(\displaystyle \frac{1}{2}ab\cdot\sin C\), but Im not sure.

          B
          *
        / :
      a/  :h
      /   :
   C* - - - - - *A
          b

Consider \(\displaystyle \Delta ABC\) . . . with side BC pivoting about vertex C.
\(\displaystyle \;\;\;\)Its area is: \(\displaystyle \,A\;=\;\frac{1}{2}bh\)

Since \(\displaystyle a\) and \(\displaystyle b\) is fixed, the area is a maximum when \(\displaystyle h\) is a maximum.
\(\displaystyle \;\;\;\)This occurs when \(\displaystyle h\,=\,a\) . . . that is, \(\displaystyle \angle C\,=\,90^o\).

[The area is a minimum when \(\displaystyle h\,=\,0\) . . . when \(\displaystyle c\,=\,0^o\) or \(\displaystyle 180^o\).]


I think your approach is even better: \(\displaystyle \:A\:=\:\frac{1}{2}ab\cdot\sin C\)

Since \(\displaystyle a\) and \(\displaystyle b\) are fixed,
\(\displaystyle \;\;\;\) the maximum area occurs when: \(\displaystyle \sin C = 1\;\;\Rightarrow\;\;C\,=\,90^o.\)

 

[xc630]

so the minimum would be when the angle is 0 degrees? but wouldnt this mean that this isnt even a triangle?

 

[pka]

I really disagree that a triangle can have an angle of 0!
If the triangle cannot have an angle of 0 then it cannot have area 0.
Therefore, the best we can do is to say that the minimum approaches 0.

 

[stapel]

I really disagree that a triangle can have an angle of 0!

Might this be what is terms a "degenerate" triangle...?

Eliz.