21．解：

（1）∵DE⊥AB，∴?DEA?90?

又∵?DAB?45，∴DE?AE． ······································································· （1分） 在Rt△DEB中，?DEB?90?，tanB?3DE3?． ·，∴······························ （1分）

4BE4设DE?3x，那么AE?3x，BE?4x．

∵AB?7，∴3x?4x?7，解得x?1． ······························································ （2分） ∴DE?3． ············································································································· （1分） （2） 在Rt△ADE中，由勾股定理，得AD?32． ················································ （1分）

同理得BD?5． ······································································································ （1分） 在Rt△ABC中，由tanB?∴CD?

2834，可得cosB?．∴BC?． ····················· （1分） 4553． ············································································································ （1分） 5∴cos?CDA?CD2． ················································································ （1分） ?AD102． 10

即?CDA的余弦值为青浦区

21. (本题满分10分，第（1）、（2）小题，每小题5分）

如图5，在Rt△ABC中，∠C=90°，AC=3，BC=4，∠ABC的平分线交边AC于点D，延长BD至点E，且BD=2DE，联结AE． （1）求线段CD的长； （2）求△ADE的面积.

21．解：（1）过点D作DH⊥AB，垂足为点H． ································································· （1分）

∵BD平分∠ABC，∠C=90°，

∴DH = DC=x， ··································································································· （1分）

BDCAE图5

则AD=3?x．

∵∠C=90°，AC=3，BC=4，∴AB=5. ····························································· （1分） ∵sin?BAC?∴

HDBC?， ADABx4?， ·································································································· （1分） 3?x54∴x?. ·········································································································· （1分）

311410（2）SABD?AB?DH??5??. ···························································· （1分）

2233∵BD=2DE， ∴

SSABDADE??BD···················································································· （3分） ?2， ·

DE1015??. ···················································································· （1分） 323∴SADE松江区

21.（本题满分10分, 每小题各5分） 如图，已知△ABC中，∠B=45°，tanC?BC=6.

（1）求△ABC面积；

（2）AC的垂直平分线交AC于点D，交BC于 点E. 求DE的长．

21.（本题满分10分, 每小题各5分）

解：（1）过点A作AH⊥BC于点H…………1分 在Rt?ABC中，∠B=45°

设AH =x ,则BH=x………………………………1分

(第21题图)

1， 2A D B (第21题图) E C A

D E C 在Rt?AHC中，tanC?AH1? HC2∴HC=2x………………………………………………………1分 ∵BC=6

∴x+2x=6 得x=2

∴AH=2…………………………………………………………1分 1∴S?ABC??BC?AH?6……………………………………1分

2(2)由（1）得AH=2，CH=4

在Rt?AHC中，AC?AH2?HC2?25…………………2分 ∵DE垂直平分AC ∴CD?1AC?5 2 ED⊥AC …………………………………………………1分 在Rt?EDC中，tanC?∴DE?ED1?……………………………1分 CD215 ………………………………………………1分 2徐汇区

21. 如图，在Rt?ABC中，?C?90?，AC?3，BC?4，AD平分?BAC交BC于点D. （1）求tan?DAB；

（2）若⊙O过A、D两点，且点O在边AB上，用 尺规作图的方法确定点O的位置并求出的⊙O半径. （保留作图轨迹，不写作法）

杨浦区

21、（本题满分10分，第（1）小题满分3分，第（2）小题满分7分）

已知，如图5，在梯形ABCD中，DC//AB, AD=BC, BD平分∠ABC，∠A=600 求：（1）求∠CDB的度数

（2）当AD=2时，求对角线BD的长和梯形ABCD的面积。