10.
\(\Delta ABC\)中，\(\overline{AB}=5\)，\(\overline{AC}=3\)，\(∠A=2∠B\)，則\(\Delta ABC\)之內切圓半徑為 。
[解答]
做\(∠A\)之角平分線\(\overline{AD}\)交\(\overline{BC}\)於\(D\)，設\(\overline{BD}=5x,\overline{DC}=3x\)
由\(\Delta ABC \cong \Delta DAC\)，得\(\displaystyle \frac{3x}{3}=\frac{3}{8x}\Rightarrow x=\frac{\sqrt{6}}{4}\)
\(\displaystyle cosA=\frac{5^2+3^2-(2\sqrt{6})^2}{2 \times 3 \times 5}=\frac{1}{3}\Rightarrow sinA=\frac{2\sqrt{2}}{3}\)
再由\(\displaystyle \Delta =\frac{1}{2}\times 3 \times 5 \times \frac{2\sqrt{2}}{3}=rs=r \times(4+\sqrt{6})\Rightarrow r=2\sqrt{2}-3\)，其中\(\displaystyle s=\frac{5+3+2\sqrt{6}}{2}\)
