Thread: Optimization problem of a triangle inscribed in a circle

1. Optimization problem of a triangle inscribed in a circle

I need your help understanding how to go about the following problem,
Triangle ABC has sides AB = BC. It is inscribed in a circle with center O, radius 10.0 cm. Find the value of the angle BAC that produces a maximum area for triangle ABC.

2. Re: Optimization problem of a triangle inscribed in a circle

$\displaystyle a=AB=BC$

$\displaystyle b=AC$

area $\displaystyle = \frac{1}{2}a b \sin A$

use the law of sines

3. Re: Optimization problem of a triangle inscribed in a circle

I am supposed to find an equation, derivate it and then find theta. I am looking into that equation that you suggested, but i am stuck on how to draw it.

4. Re: Optimization problem of a triangle inscribed in a circle

$Area = \dfrac{1}{2}(2y)(x+10) = y(x+10)$

$\dfrac{dA}{dx} = y + (x+10) \cdot \dfrac{dy}{dx}$

$x^2+y^2=10^2 \implies \dfrac{dy}{dx} = -\dfrac{x}{y}$ ...

$\dfrac{dA}{dx} = y + (x+10) \cdot \left(-\dfrac{x}{y} \right) = y - \dfrac{x^2}{y} - \dfrac{10x}{y}$

$y - \dfrac{x^2}{y} - \dfrac{10x}{y} = 0$

$y^2 - x^2 - 10x = 0$

$(10^2 - x^2) - x^2 - 10x = 0$

$2x^2 + 10x - 100 = 0 \implies 2(x+10)(x-5) = 0 \implies x = 5$

$m\angle{A} = \arctan\left(\dfrac{10+x}{y}\right)$

5. Re: Optimization problem of a triangle inscribed in a circle

Thank you for your help. There is an easier way of solving it. If I put the sides in terms of sine and cosine and then differentiate. Your drawing is very helpful. ��

6. Re: Optimization problem of a triangle inscribed in a circle

Originally Posted by Dayli
Thank you for your help. There is an easier way of solving it. If I put the sides in terms of sine and cosine and then differentiate. Your drawing is very helpful. ��
Using Skeeter's lovely diagram Area$(\Delta BCM)=A=0.5(x+10)(y)=0.5(x+10)(\sqrt{100-x^2})$
You should be able to maximize $A$. Find the $m(\angle CBM)$. Double it to find $m(\angle CBA)$

7. Re: Optimization problem of a triangle inscribed in a circle

Originally Posted by Dayli
I am supposed to find an equation, derivate it and then find theta. I am looking into that equation that you suggested, but i am stuck on how to draw it.
$\displaystyle R=10$

$\displaystyle a=2R \sin A$ and $\displaystyle b=2R \sin B=2R \sin (2A)$ since $\displaystyle B=180-2A$

Therefore the area is given by

$\displaystyle \frac{1}{2}(2R \sin A)* 2R \sin (2A)*\sin A$

$\displaystyle =2R^2 \sin ^2A *\sin (2A)$

differentiate and solve for $\displaystyle A$