Javob: ... \( \mathrm{cm}^{2} \). 10. Romb burchaklarining nisbati \( 1: 5 \mathrm{ga} \), tomoni esa a ga teng. Shu rombning yuzini toping.

Rombning yuzini topish uchun avval rombning burchaklarini va tomonini bilishimiz kerak. Rombning burchaklari nisbati \(1:5\) bo'lsa, burchaklarni quyidagicha belgilaymiz: - \( \alpha \) - kichik burchak - \( \beta \) - katta burchak Burchaklar nisbati \(1:5\) bo'lgani uchun, \( \beta = 5\alpha \) deb yozishimiz mumkin. Rombning ichki burchaklari yig'indisi \(360^\circ\) ga teng, shuning uchun: \[ \alpha + \beta + \alpha + \beta = 360^\circ \] Bu tenglamani soddalashtirsak: \[ 2\alpha + 2\beta = 360^\circ \] Bu yerda \( \beta = 5\alpha \) ni o'rnatamiz: \[ 2\alpha + 2(5\alpha) = 360^\circ \] Bu tenglamani yechamiz: \[ 2\alpha + 10\alpha = 360^\circ \] \[ 12\alpha = 360^\circ \] \[ \alpha = 30^\circ \] Shunday qilib, \( \beta \) ni topamiz: \[ \beta = 5\alpha = 5 \times 30^\circ = 150^\circ \] Endi rombning yuzini topish uchun quyidagi formuladan foydalanamiz: \[ S = a^2 \cdot \sin(\alpha) \] Bu yerda \( \alpha = 30^\circ \) va \( a \) rombning tomoni. \( \sin(30^\circ) = \frac{1}{2} \) ekanligini bilamiz. Shuning uchun: \[ S = a^2 \cdot \sin(30^\circ) = a^2 \cdot \frac{1}{2} = \frac{a^2}{2} \] Natijada rombning yuzi: \[ S = \frac{a^2}{2} \, \mathrm{cm}^{2} \] Shunday qilib, rombning yuzi \( \frac{a^2}{2} \, \mathrm{cm}^{2} \) ga teng.