《2025年天府前沿九年级数学上册北师大版》

7. (2025·东部新区)如图,若D,E分别为△ABC中AB,AC边上的点,且∠AED= ∠B,AD= 3,AC= 6,DB= 5,则AE的长为 ()

A. 9/4
B. 5/2
C. 18/5
D. 4

答案： D

8. (2025·武侯)如图,在平面直角坐标系xOy中,矩形OABC的顶点坐标分别是O(0,0),A(6,0),B(6,4),C(0,4).已知矩形OA'B'C'与矩形OABC位似,位似中心是原点O,且矩形OA'B'C'的面积等于矩形OABC的面积的1/4,则点B'的坐标是 ()

A. (3,2)
B. (-3,-2)
C. (3,2)或(-3,-2)
D. (2,3)或(-2,-3)

答案： C

9. (2025·编写)如图,在矩形ABCD中,E是边AB的中点,连接DE交对角线AC于点F.若AB= 4,AD= 3,求CF的长.

答案： 【解】
∵四边形$ABCD$为矩形，$\therefore AB = CD$，$AD = BC$，$AB // CD$，$\therefore \angle FAE = \angle FCD$。又$\because \angle AFE = \angle CFD$，$AB = 4$，$AE = BE$，$\therefore \triangle AFE \backsim \triangle CFD$，$\therefore \frac{CF}{AF} = \frac{CD}{AE} = 2$。$\because AB = 4$，$AD = BC = 3$，$\therefore AC = \sqrt{AB^{2} + BC^{2}} = 5$，$\therefore CF = \frac{2}{3}AC = \frac{10}{3}$。

10. (2022·郫都)如图,过矩形ABCD(AD＞AB)的对角线AC的中点O作AC的垂直平分线EF,分别交AD,BC于点E,F,分别连接AF和CE.
(1)求证:四边形AFCE是菱形;
(2)过点E作AD的垂线交AC于点P,求证$:2AE^2= AC·AP.$

答案： 【证明】
(1)
∵$EF$垂直平分$AC$，$\therefore AO = CO$，$\angle AOE = \angle COF = 90^{\circ}$，$EA = EC$，$FA = FC$。
∵四边形$ABCD$为矩形，$\therefore AD // BC$，$\therefore \angle EAO = \angle FCO$。
在$\triangle AOE$和$\triangle COF$中，$\begin{cases} \angle EAO = \angle FCO \\ AO = CO \\ \angle AOE = \angle COF \end{cases}$，$\therefore \triangle AOE \cong \triangle COF(ASA)$，$\therefore AE = CF$，$\therefore AE = EC = AF = CF$，$\therefore$四边形$AFCE$是菱形。
(2)
∵$AE \perp EP$，$\therefore \angle AEP = \angle AOE = 90^{\circ}$。在$\triangle AEP$和$\triangle AOE$中，$\angle PAE = \angle EAO$，$\angle AEP = \angle AOE$，$\therefore \triangle AEP \backsim \triangle AOE$，$\therefore \frac{AE}{AP} = \frac{AO}{AE}$，$\therefore AE^{2} = AO \cdot AP$。$\because AO = \frac{1}{2}AC$，$\therefore AE^{2} = \frac{1}{2}AC \cdot AP$，$\therefore 2AE^{2} = AC \cdot AP$。

11. (2022·温江)如图,已知菱形ABCD中,G是对角线BD上一点,延长CG分别交边AD和BA的延长线于点E,F.若GE= 2,EF= 5,则CG的长为______.

答案： $\sqrt{14}$

12. (2025·编写)已知P是边长为4的正方形ABCD内一点,且PB= 3,BF⊥BP,垂足是B.若在射线BF上找一点M,使以B,M,C为顶点的三角形与△ABP相似,则BM为______.

答案： 3或$\frac{16}{3}$

13. (2025·青羊)如图,在平面直角坐标系xOy中,一次函数y = 1/2x + 2的图象分别交x轴,y轴于A,B两点,过该函数图象上一点C(4,4)作CD⊥x轴于点D,E是线段AB上一动点,连接BD,EO.若以B,E,O为顶点的三角形与△BCD相似,则点E的坐标为______.

答案： $\left(-\frac{8}{5}, \frac{6}{5}\right)$或$(-2,1)$