23．（本题8分）
解：设电子警察安装在悬臂灯杆上的高度AD的长为x m．
在Rt△ADB中，tan∠ABD＝
AD ， ········································································· 1分 BD
ADx
∴ BD＝＝ ． ················································································· 2分
tan∠ABDtan18°在Rt△ACD中，tan∠ACD＝
AD
， ··········································································· 3分 CD
ADx
∴ CD＝＝ ． ················································································· 4分
tan∠ACDtan14°∵ BC＝CD－BD， ∴
xx
－＝6． tan14°tan18°
40
∴ 4x－x＝6． ·································································································· 6分
13解这个方程，得x＝6.5． ······················································································· 7分 答：电子警察安装在悬臂灯杆上的高度AD的长为6.5 m． ············································ 8分
24．（本题8分）
解：设每个小组有学生x名． ························································································ 1分
240240
根据题意，得－＝4．··················································································· 4分
2x3x解这个方程，得x＝10． ························································································ 6分 经检验，x＝10是原方程的根． ··············································································· 7分 答：每个小组有学生10名．··················································································· 8分 （说明：如果学生只设了未知数，没有用未知数表示相关量不给分）
25．（本题8分）
解：（1）证明：∵BE＝BC，
∴∠BEC＝∠BCE． ······································ 1分 ∵四边形ABCD是平行四边形， ∴AD∥BC，AB∥CD．
∴∠BCE＝∠DEC，∠A＋∠D＝180°．
∴∠BEC＝∠DEC． ····················································································· 2分 ·∵四边形ABCD内接于⊙O， ∴∠A＋∠BCE＝180°．
∴∠BCE＝∠D． ·························································································· 3分 ∴△BEC∽△CED． ······················································································ 4分 （2）过点O作OF⊥CE，垂足为F，连接OC． 1
∴CF＝CE． ······························································································ 5分
2
页（共5页） 数学答案 第9
B O F C A E D
∴直线OF垂直平分CE． ∵BE＝BC，
∴直线OF经过点B．
∵△BEC∽△CED，又由（1）可知CE＝CD， BCCE∴＝． CEDE
∵BC＝10，DE＝3.6，
∴CE＝CD＝6． ··························································································· 6分 1
∴CF＝CE＝3．
2设⊙O的半径为r．
易得BF＝BC2－CF2＝91，OF＝91－r． 在Rt△OCF中，OF2＋CF2＝OC2，
∴(91－r)2＋9＝r2． ···················································································· 7分 5091∴r＝． ······························································································ 8分
91
26．（本题9分）
解：（1）s1、s2与x之间的函数图像如图所示．
O s2 s1 s/km 900 4 6 12 x/km
····································· 4分
（2）根据题意，得s1＝900－75x． ············································································· 5分
当x＝4.5时，s1＝562.5，
设s3与x之间的函数表达式为s3＝150x＋b． 当x＝4.5时，s3＝562.5，
s3＝150x－112.5． ···························································································· 7分 （3）根据题意，当s3＝0时，x＝0.75． ······································································· 8分
所以第二列快车比第一列快车晚出发0.75小时．···················································· 9分
页（共5页） 数学答案 第10
27．（本题11分）
解：（1）如图，△DEF即为所求．
B C D F
A E ····································· 2分
（说明：不写结论扣1分）
（2）①设△ABC的姊妹三角形为△DEF，且DE＝DF．
∵在△ABC中，AB＝AC，∠A＝30°，BC＝6－2， ∴∠B＝∠C＝75°．
过点B作BG⊥AC，垂足为G．设BG＝x， 则AB＝AC＝2x，AG＝3x．
∴CG＝AC－AG＝2x－3x＝(2－3)x． 在Rt△BGC中，BG2＋CG2＝BC2， ∴x2＋(2－3)2x2＝(6－2)2． ∴x＝1．
A G B C ∴AB＝AC＝2． ····························································································· 3分 第一种情形：∠D＝∠ABC＝75°， ····································································· 4分 DE＝DF＝BC＝6－2． ················································································ 5分 第二种情形：当∠E＝∠A＝30°时，∠EDF＝120°． ·············································· 6分 EF＝AB＝2．
过点D作DH⊥EF，垂足为H． 1
∵DE＝DF，∴EH＝EF＝1．
2∴ED＝
EH23＝． cos30°3
E
H
F
D 23∴△ABC的姊妹三角形的顶角为75°时，腰长为6－2；顶角为120°时，腰长为．
3 ·················································································································· 7分 ②36． ········································································································· 9分 （3）①③． ········································································································ 11分
页（共5页） 数学答案 第11
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