Giải câu 3 trang 127 toán VNEN 9 tập 1.
a) Ta có:
BC = $\sqrt{(R + R')^{2} - (R - R')^{2}}$ = $\sqrt{R^{2} + 2RR' + R'^{2} - R^{2} + 2RR' - 2R'^{2}}$ = $\sqrt{4RR'}$ = 2$\sqrt{RR'}$
b) Ta có:
MB = $\sqrt{(R + r)^{2} - (R - r)^{2}}$ = 2$\sqrt{Rr}$
MC = $\sqrt{(R' + r)^{2} - (R' - r)^{2}}$ = 2$\sqrt{R'r}$
MB + MC = BC
$\Rightarrow $ 2$\sqrt{Rr}$ + 2$\sqrt{R'r}$ = 2$\sqrt{RR'}$
$\Leftrightarrow $ $\sqrt{Rr}$ + $\sqrt{R'r}$ = $\sqrt{RR'}$
$\Leftrightarrow $ $\sqrt{r}$ = $\frac{\sqrt{RR'}}{\sqrt{R} + \sqrt{R'}}$
$\Rightarrow $ r = $\frac{RR'}{(\sqrt{R} + \sqrt{R'})^{2}}$.
Vậy r = $\frac{RR'}{(\sqrt{R} + \sqrt{R'})^{2}}$.