The figure below shows two intersect v radii 9cm with centers O, on ① Find the length of the common chord AB II The area common to the 2 circles

Tuna radii $r_1 = 4$ cm na $r_2 = 9$ cm.
Umbali kati ya vituo $O_1O_2 = 9$ cm (kwa dhana).
Acha $M$ iwe sehemu ya makutano ya kodi ya kawaida AB na mstari unaounganisha vituo $O_1O_2$.
Kodi ya kawaida AB imegawanywa katika sehemu mbili sawa na $M$, hivyo $AM = MB$. Acha $AM = h$.
Acha $O_1M = x$. Kisha $O_2M = O_1O_2 - x = 9 - x$.
Katika pembetatu ya kulia $\triangle O_1MA$:

Katika pembetatu ya kulia $\triangle O_2MA$:

Linganisha milinganyo (1) na (2):

Badilisha thamani ya $x$ katika mlinganyo (1):

Urefu wa kodi ya kawaida AB ni $2h$:

Urefu wa kodi ya kawaida AB ni $\boxed{\frac{8\sqrt{77}}{9} \text{ cm}}$.

II) Tafuta eneo la kawaida la duara mbili.

Eneo la kawaida ni jumla ya sehemu mbili za duara. Eneo la sehemu ya duara = Eneo la sekta - Eneo la pembetatu.

$O_1M = x = \frac{8}{9}$ cm.
$\cos(\angle AO_1M) = \frac{O_1M}{r_1} = \frac{8/9}{4} = \frac{2}{9}$.
Pembe ya sekta $\theta_1 = \angle AO_1B = 2 \angle AO_1M = 2 \arccos\left(\frac{2}{9}\right)$ radiani.
Eneo la sekta $O_1AB = \frac{1}{2} r_1^2 \theta_1 = \frac{1}{2} (4^2) \left(2 \arccos\left(\frac{2}{9}\right)\right) = 16 \arccos\left(\frac{2}{9}\right) \text{ cm}^2$.
Eneo la pembetatu $\triangle O_1AB = \frac{1}{2} \times AB \times O_1M = \frac{1}{2} \times \left(\frac{8\sqrt{77}}{9}\right) \times \left(\frac{8}{9}\right) = \frac{32\sqrt{77}}{81} \text{ cm}^2$.
Eneo la sehemu ya kwanza = $16 \arccos\left(\frac{2}{9}\right) - \frac{32\sqrt{77}}{81} \text{ cm}^2$.

$O_2M = 9 - x = 9 - \frac{8}{9} = \frac{81-8}{9} = \frac{73}{9}$ cm.
$\cos(\angle AO_2M) = \frac{O_2M}{r_2} = \frac{73/9}{9} = \frac{73}{81}$.
Pembe ya sekta $\theta_2 = \angle AO_2B = 2 \angle AO_2M = 2 \arccos\left(\frac{73}{81}\right)$ radiani.
Eneo la sekta $O_2AB = \frac{1}{2} r_2^2 \theta_2 = \frac{1}{2} (9^2) \left(2 \arccos\left(\frac{73}{81}\right)\right) = 81 \arccos\left(\frac{73}{81}\right) \text{ cm}^2$.
Eneo la pembetatu $\triangle O_2AB = \frac{1}{2} \times AB \times O_2M = \frac{1}{2} \times \left(\frac{8\sqrt{77}}{9}\right) \times \left(\frac{73}{9}\right) = \frac{292\sqrt{77}}{81} \text{ cm}^2$.
Eneo la sehemu ya pili = $81 \arccos\left(\frac{73}{81}\right) - \frac{292\sqrt{77}}{81} \text{ cm}^2$.

Eneo la kawaida = Eneo la sehemu ya kwanza + Eneo la sehemu ya pili

Eneo la kawaida la duara mbili ni $\boxed{16 \arccos\left(\frac{2}{9}\right) + 81 \arccos\left(\frac{73}{81}\right) - 4\sqrt{77} \text{ cm}^2}$.