# Optimizing the Area of a Triangle

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DMac
[SOLVED] Optimizing the Area of a Triangle

I've been doing countless other ones, but this one has really stumped me.

"Triangle ABC has AB = AC. It is inscribed in a circle centre O, radius 10 cm. Find the value of the angle BAC that produces a maximum area for the triangle ABC."

I've drawn lots of diagrams to help me get started, but to no avail. I seriously have no clue how to do this one. Could someone please tell me a clue as to what my first step should be? I just need a starting point. Thanks in advance.

DMac
P.S. I'm not looking for an actual answer, I just need a pointer as to how to approach this problem.

durt
Here's one approach. You can try to find the area $$K$$ of the triangle in terms of $$r$$ (10 cm) and $$\theta$$, which is half of $$\angle BAC$$. Then you can just use some calculus on this. To find this relationship I would let the length base of the triangle ($$BC$$) be $$b$$ and the height $$h$$. Now here's a hint: let $$M$$ be the midpoint of $$BC$$. What do you know about $$\triangle OMC$$?

This is a little tedious. The area of the triangle inscribed in a circle is given by $\frac{1}{2}(AB)^2sin \theta$. Let x=AB for convenience of notation here. You know how to find the maximum area given the above expression. But x here isn't a constant, so you can't differentiate the function and treat it like one. Instead, if you draw the diagram, you'll find a way to express x in terms of BC. But again BC is also a variable, just like x, since varying [itex]\theta[/tex] will also change BC. But by a certain angular property of circles, you can express BC in terms of r (the radius which is constant) and [itex]\theta[/tex]. Then once you have this expression, substitute back into the expression for the area. Now you can differentiate it properly.