《2025年大联考单元期末测试卷九年级数学全一册沪科版》

14. 如图,在四边形$ABCD$中,$AC$与$BD$相交于点$O$,$\angle ABC = \angle DAC = 90^{\circ}$,$\tan \angle ACB = \frac{1}{2}$,$\frac{BO}{OD} = \frac{4}{3}$.
(1)$\frac{CO}{AO} =$$\frac{22}{3}$.
(2)$\frac{S_{\triangle ABD}}{S_{\triangle CBD}} =$$\frac{3}{32}$.

答案： 14.
(1)$\frac{22}{3}$
(2)$\frac{3}{32}$

15. 计算:$\frac{\sin 30^{\circ}}{\cos 45^{\circ}} - \sqrt{(1 - \tan 30^{\circ})^{2}} - \tan 45^{\circ}$.

答案： 15.解:原式=$\frac{\frac{1}{2}}{\frac{\sqrt{2}}{2}\sqrt{1-(\frac{\sqrt{3}}{3})^2}}-1$(3分)
=$\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}(1-\frac{\sqrt{3}}{3})}-1$
=$\frac{\sqrt{2}}{2}-1+\frac{\sqrt{3}}{3}-1$
=$\frac{\sqrt{3}}{3}+\frac{\sqrt{2}}{2}-2$.(8分)

16. 在$Rt\triangle ABC$中,$\angle C = 90^{\circ}$,$\angle A$,$\angle B$,$\angle C$的对边分别为$a$,$b$,$c$,请你确定式子$\frac{a^{2}}{bc}\cos A + \frac{b^{2}}{ac}\cos B$是否为常数,并试证明你的结论.

答案： 16.解:$\frac{a^2}{bc}\cos A+\frac{b^2}{ac}\cos B$是常数.(1分)
证明:$\because \cos A=\frac{b}{c},\cos B=\frac{a}{c},a^2+b^2=c^2$,
$\therefore \frac{a^2}{bc}\cos A+\frac{b^2}{ac}\cos B$
=$\frac{a^2}{bc}· \frac{b}{c}+\frac{b^2}{ac}· \frac{a}{c}$(5分)
=$\frac{a^2}{c^2}+\frac{b^2}{c^2}$
=$\frac{a^2 + b^2}{c^2}$
=$\frac{c^2}{c^2}$
=1,
$\therefore \frac{a^2}{bc}\cos A+\frac{b^2}{ac}\cos B$是常数.(8分)