《2025年学习检测九年级数学上册华师大版河南专版》

13. 如图,在$\triangle ABC$中,底边$BC = 60cm$,高$AD = 40cm$,四边形$SPQR$是矩形,且$SR:SP = 1:2$,求矩形$SPQR$的面积.

答案： $450\ cm^2$.

14. 如图,在$□ ABCD$中,点$E在AD$的延长线上,$BE与CD相交于点F$.
(1) 求证：$\triangle ABE\backsim\triangle CFB$；
(2) 若$\triangle DEF的面积为4$,$\frac{DF}{CF} = \frac{2}{3}$,求$\triangle ABE$的面积.

答案：
(1)
∵ 四边形ABCD 是平行四边形，$\therefore AD// BC,\angle A=\angle C$.$\therefore \angle CBE=\angle E$.$\therefore \triangle ABE\backsim \triangle CFB$.
(2)
∵ 四边形ABCD 是平行四边形，$\therefore AB// CD,AB=CD$.$\therefore \triangle DEF\backsim \triangle AEB$.$\because \frac {DF}{CF}=\frac {2}{3}$,$\therefore \frac {DF}{CD}=\frac {DF}{AB}=\frac {2}{5}$.$\therefore S_{\triangle DEF}:S_{\triangle AEB}=(\frac {2}{5})^{2}=4:25$.$\because S_{\triangle DEF}=4$,$\therefore S_{\triangle ABE}=25$.