Inscribing a rectangle problem

[ticaaal70]

Inscribe a rectangle of base B and height H and an isosceles triangle of base B in a circle of radius one as shown.
For what value of H do the rectangle and triangle have the same area?



Thanks for the help

 

[stapel]

Inscribe a rectangle of base B and height H and an isosceles triangle of base B in a circle of radius one as shown. For what value of H do the rectangle and triangle have the same area?

What have you tried? How far have you gotten? Where are you stuck?

Please be complete. Thank you!

 

[edumat]

$\displaystyle h = 2sen(x)$
$\displaystyle b = 2cos(x)$
Can you see this?
Now:
$\displaystyle A_{rect} = ?$
$\displaystyle A_{triang} = ?$

 

[DrPhil]

View attachment 3173
$\displaystyle h = 2sen(x)$
$\displaystyle b = 2cos(x)$
Can you see this?
Now:
$\displaystyle A_{rect} = ?$
$\displaystyle A_{triang} = ?$

The rectangle is straightforward:
$\displaystyle A_{rect} = h \times b = 4\ \sin(x)\ \cos(x)$

The base of the triangle is $\displaystyle b$ and its altitude is $\displaystyle (1 - h/2)$
$\displaystyle A_{triang} = \frac{1}{2}\ b \ (1-h/2) = \cos(x)\ \left(1 - \sin(x)\right)$

I might try making $\displaystyle h$ the independent variable (instead of $\displaystyle x$), and make the RATIO of the two areas equal to 1. Won't $\displaystyle b$ cancel out if you take a ratio?