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.