《2025年全程检测单元测试卷九年级数学全一册湘教版》

17. (12分)如图6,在$\triangle ABC$中,$AB=AC$,$BD\perp AC$于点$D$,$AC=15$,$\sin A=\frac{3}{5}$.
(1)(6分)求$BD$的长.
(2)(6分)求$\tan C$的值.

答案： 17．
(1)$\because AB=AC, AC=15, \therefore AB=15$. $\because BD \perp AC, \therefore \angle ADB=\angle BDC=90°$. $\therefore$在$\mathrm{Rt}\triangle ADB$中，$\sin A=\frac{BD}{AB}=\frac{3}{5}. \therefore \frac{BD}{15}=\frac{3}{5}. \therefore BD=9$.
(2)$\because$在$\mathrm{Rt}\triangle ADB$中，$\angle ADB=90°, AB=15, BD=9, \therefore AD=\sqrt{AB^2-BD^2}=12. \therefore CD=AC-AD=15-12=3$. $\therefore$在$\mathrm{Rt}\triangle BCD$中，$\tan C=\frac{BD}{CD}=\frac{9}{3}=3$.