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?