第27章 圆 学情评估卷（含答案）华东师大版数学九年级下册
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| 名称 | 第27章 圆 学情评估卷（含答案）华东师大版数学九年级下册 |  | |
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| 科目 | 数学 | ||
| 更新时间 | 2025-09-12 15:12:44 | ||
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第27章 学情评估卷
一、选择题(每题3分，共30分)
1. 若⊙O的半径为4 cm，点A到圆心O的距离为3 cm，则点A与⊙O的位置关系是( )
A．点A在⊙O内 B．点A在⊙O上
C．点A在⊙O外 D．不能确定
2．如图，在⊙O中，＝，∠1＝45°，则∠2等于( )
A．60° B．30° C．45° D．40°
INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\26春9HSJ-9.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\26春9HSJ-9.tif" \* MERGEFORMATINET INCLUDEPICTURE "26春9HSJ-9.tif" \* MERGEFORMAT INCLUDEPICTURE "26春9HSJ-9.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\九SJ-4.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\九SJ-4.tif" \* MERGEFORMATINET INCLUDEPICTURE "九SJ-4.tif" \* MERGEFORMAT INCLUDEPICTURE "九SJ-4.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\9数HSJ-12.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\9数HSJ-12.tif" \* MERGEFORMATINET INCLUDEPICTURE "9数HSJ-12.tif" \* MERGEFORMAT INCLUDEPICTURE "9数HSJ-12.tif" \* MERGEFORMAT
(第2题) (第3题) (第4题)
3．[教材P56习题T9变式]如图，PA，PB分别切⊙O于A，B两点，如果∠P＝60°，PA＝2，那么△PAB的周长为( )
A．2 B．6 C．8 D．4
4．如图，在墙壁中埋着一根半径未知的圆柱形木材，现用锯子去锯这根木材，锯口深CD＝4，锯道AB＝16，则该木材的半径是( )
A．20 B．12 C．10 D．8
5．下列说法正确的是( )
A．三点确定一个圆
B．经过圆心的直线是圆的对称轴
C．和半径垂直的直线是圆的切线
D．三角形的内心到三角形三个顶点的距离相等
6．如图，在Rt△ABC中，∠ACB＝90°，∠ABC＝40°，AB＝6，斜边AB是半圆O的直径，点D是半圆上的一个动点，连结CD与AB交于点E，当BE＝BC时，弧BD的长为( )
A.π B.π C.π D.π
INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\9数HSJ-14.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\9数HSJ-14.tif" \* MERGEFORMATINET INCLUDEPICTURE "9数HSJ-14.tif" \* MERGEFORMAT INCLUDEPICTURE "9数HSJ-14.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\9数HSJ-15.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\9数HSJ-15.tif" \* MERGEFORMATINET INCLUDEPICTURE "9数HSJ-15.tif" \* MERGEFORMAT INCLUDEPICTURE "9数HSJ-15.tif" \* MERGEFORMAT
(第6题) (第7题)
7．如图，AB是⊙O的切线，B为切点，连结AO交⊙O于点C，延长AO交⊙O于点D，连结BD，若∠A＝2∠D，且AB＝4，则AC的长是( )
A．1 B. C．4－2 D．4 －4
8. 如图，点A、B、C、D都在⊙O上，∠C＝120°，AB＝AD＝8，则点O到BD的距离是( )
A. B. C．3 D．4
INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\9数HSJ-16.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\9数HSJ-16.tif" \* MERGEFORMATINET INCLUDEPICTURE "9数HSJ-16.tif" \* MERGEFORMAT INCLUDEPICTURE "9数HSJ-16.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\26春9HSJ-10.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\26春9HSJ-10.tif" \* MERGEFORMATINET INCLUDEPICTURE "26春9HSJ-10.tif" \* MERGEFORMAT INCLUDEPICTURE "26春9HSJ-10.tif" \* MERGEFORMAT
(第8题) (第9题)
9．如图，正八边形ABCDEFGH内接于⊙O，连结AE，HE，若S△AEH＝4，则⊙O的半径为( )
A．2 B．2 C．2 D．4
10. 如图，在菱形ABCD中，AB＝2，∠B＝120°，点O是对角线AC的中点，以点O为圆心，OA长为半径作圆心角为60°的扇形OEF，点D在扇形OEF内，则图中阴影部分的面积为( )
A.－ B．π－
C.－ D．无法确定
二、填空题(每题3分，共15分)
11. 一个圆锥的母线长为6，底面圆的半径为4，那么这个圆锥的侧面积是________. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
12．如图，AB是⊙O的直径，C，D是⊙O上的两点，连结BC，CD，AD，BD，若∠ADC＝25°，BC平分∠ABD，则∠BAD的度数为________. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\9HSJ+1.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\9HSJ+1.tif" \* MERGEFORMATINET INCLUDEPICTURE "9HSJ+1.tif" \* MERGEFORMAT INCLUDEPICTURE "9HSJ+1.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\一改-1.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\一改-1.tif" \* MERGEFORMATINET INCLUDEPICTURE "一改-1.tif" \* MERGEFORMAT INCLUDEPICTURE "一改-1.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\9数HSJ-23.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\9数HSJ-23.tif" \* MERGEFORMATINET INCLUDEPICTURE "9数HSJ-23.tif" \* MERGEFORMAT INCLUDEPICTURE "9数HSJ-23.tif" \* MERGEFORMAT
(第12题) (第13题) (第14题)
13. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\情境题门洞改造.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\情境题门洞改造.tif" \* MERGEFORMATINET INCLUDEPICTURE "情境题门洞改造.tif" \* MERGEFORMAT INCLUDEPICTURE "情境题门洞改造.tif" \* MERGEFORMAT 如图，墙上有一个矩形门洞，宽1.5 m，高2 m，现要将其改造成圆弧形门洞，则改造后圆弧形门洞的最大高度是__________m. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
14. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\新考法新函数思想.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\新考法新函数思想.tif" \* MERGEFORMATINET INCLUDEPICTURE "新考法新函数思想.tif" \* MERGEFORMAT INCLUDEPICTURE "新考法新函数思想.tif" \* MERGEFORMAT 如图，在平面直角坐标系xOy中，若动点P在抛物线y＝ax2上，⊙P恒过点F(0，2)，且与直线y＝－2始终保持相切，则a＝________. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
15．如图，AB＝3，点C是平面内一动点，且BC＝2，连结AC，将AC绕点A逆时针旋转90°，得到AD，连结BD，则BD的最小值为 ________，最大值为 ________． INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
三、解答题(本大题共8个小题，满分75分)
16. (8分)如图，在⊙O中，弦AB与CD相交于点E，AB＝CD.求证： INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
(1)＝；
(2)AE＝CE.
INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\27-15.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\27-15.tif" \* MERGEFORMATINET INCLUDEPICTURE "27-15.tif" \* MERGEFORMAT INCLUDEPICTURE "27-15.tif" \* MERGEFORMAT
17．(9分)如图，AB是半圆O的直径，C，D是半圆O上两点，连结AC，BC，OD，AD，OD∥BC，OD与AC交于点E. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
(1)若∠D＝70°，求∠CAD的度数；
(2)若AC＝8，DE＝2，求AB的长．
INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\27-17.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\27-17.tif" \* MERGEFORMATINET INCLUDEPICTURE "27-17.tif" \* MERGEFORMAT INCLUDEPICTURE "27-17.tif" \* MERGEFORMAT
18．(9分)如图，方格纸上每个小正方形的边长均为1个单位长度，点O，A，B，C均在格点(两条网格线的交点叫格点)上，以点O为原点建立平面直角坐标系. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
(1)过A，B，C三点的圆的圆心M的坐标为________；
(2)请通过计算判断点D(－3，－2)与⊙M的位置关系．
INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\9数HSJ-25.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\9数HSJ-25.tif" \* MERGEFORMATINET INCLUDEPICTURE "9数HSJ-25.tif" \* MERGEFORMAT INCLUDEPICTURE "9数HSJ-25.tif" \* MERGEFORMAT
19.(9分)小陈同学从市场上购买了如图①所示的花盆，花盆底部的横截面是直径为35 cm的圆，他家中有一个托盘，托盘底部的横截面是边长为60 cm的正三角形，如图②所示．
(1)求图②中正三角形一边的高线长；
(2)这个托盘是否适用于该花盆？请通过计算说明理由. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
20．(9分)如图，四边形ABCD内接于⊙O，连结BD，AD＝BD，延长CD至点E，使得DE＝AD，连结AE，∠ADB＝∠ADE.
(1)求证：AE∥BC；
(2)若CD＝2，DE＝6，求AB的长. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\9数HSJ-27.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\9数HSJ-27.tif" \* MERGEFORMATINET INCLUDEPICTURE "9数HSJ-27.tif" \* MERGEFORMAT INCLUDEPICTURE "9数HSJ-27.tif" \* MERGEFORMAT
21．(10分)如图，△ABC内接于⊙O，∠B＝60°.连结AO并延长，与⊙O交于点D，直线l过点C，AF⊥l，垂足为F，AC平分∠FAD，CG⊥AD，垂足为G，直线l交AD的延长线于点E.
(1)求证：直线l是⊙O的切线；
(2)若AF＝4 ，求图中阴影部分的面积. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
22. (10分) INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\情境题摩天轮.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\情境题摩天轮.tif" \* MERGEFORMATINET INCLUDEPICTURE "情境题摩天轮.tif" \* MERGEFORMAT INCLUDEPICTURE "情境题摩天轮.tif" \* MERGEFORMAT 如图①为某游乐场的摩天轮的简化示意图，将摩天轮看作直径为40米的⊙O，摩天轮上的12个轿厢看作均匀分布在⊙O上的点，乘客从点A位置上下摩天轮，摩天轮按顺时针方向匀速转动，转一周用12分钟，线段MN表示上下摩天轮的平台，OD，OE表示支架，分别过图中两个轿厢B，C的位置，OD＝OE＝24米.
(1)∠BOC＝__________°；
(2)如图②，当乘客到达点P处时，与点A之间恰好相距32米，求轿厢转过的弧AP的长度；
(3)设乘客进入轿厢t分钟时所处位置为点Q，连结DQ，若DQ与⊙O相切，直接写出此时t的值. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\26春9HSJ-14.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\26春9HSJ-14.tif" \* MERGEFORMATINET INCLUDEPICTURE "26春9HSJ-14.tif" \* MERGEFORMAT INCLUDEPICTURE "26春9HSJ-14.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\26春9HSJ-15.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\26春9HSJ-15.tif" \* MERGEFORMATINET INCLUDEPICTURE "26春9HSJ-15.tif" \* MERGEFORMAT INCLUDEPICTURE "26春9HSJ-15.tif" \* MERGEFORMAT
23. (11分) INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\新视角探究操作题.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\新视角探究操作题.tif" \* MERGEFORMATINET INCLUDEPICTURE "新视角探究操作题.tif" \* MERGEFORMAT INCLUDEPICTURE "新视角探究操作题.tif" \* MERGEFORMAT 如图，某足球场上，球员在练习射门．线段PQ表示球门，∠PAQ、∠PBQ为射门张角．理论上当射门张角越大时，进球概率越大. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
(1)如图①，∠PAQ__________∠PBQ；(填“>”“<”或“＝”)
(2)若球员沿着直线l带球跑动，如图②，以线段PQ为弦作⊙O，恰与直线l相切于点A.若点M是l上一个异于点A的动点，求证：当运动员跑动到切点A处时，射门张角最大，即∠PAQ>∠PMQ.
(3)如图③，点P(2，0)，点Q(6，0)，点A为y轴正半轴上的一个动点．
①请用尺规作图在y轴上找出点A的位置，使∠PAQ最大(保留作图痕迹，不写作法)；
②当∠PAQ最大时，请求出点A的坐标．
INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\26春9HSJ-16.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\26春9HSJ-16.tif" \* MERGEFORMATINET INCLUDEPICTURE "26春9HSJ-16.tif" \* MERGEFORMAT INCLUDEPICTURE "26春9HSJ-16.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\26春9HSJ-17.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\26春9HSJ-17.tif" \* MERGEFORMATINET INCLUDEPICTURE "26春9HSJ-17.tif" \* MERGEFORMAT INCLUDEPICTURE "26春9HSJ-17.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\26春9HSJ-18.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\26春9HSJ-18.tif" \* MERGEFORMATINET INCLUDEPICTURE "26春9HSJ-18.tif" \* MERGEFORMAT INCLUDEPICTURE "26春9HSJ-18.tif" \* MERGEFORMAT
答案
1．A 2.C 3.B 4.C 5.B 6.B 7.D 8.A 9.B 10.A 11.24π
12. 40° 13. 2.25 14. 15.4；8
16．证明：(1)∵AB＝CD，
∴＝，
∴＋＝＋，∴＝.
(2)连结AD，BC.∵＝，
∴AD＝BC.
∵＝，∴∠ADE＝∠CBE，
又∵∠AED＝∠CEB，
∴△ADE≌△CBE，∴AE＝CE.
17．解：(1)连结OC.∵OA＝OD，
∴∠OAD＝∠D＝70°，
∴∠AOD＝180°－∠OAD－∠D＝40°.
∵AB是半圆O的直径，
∴∠ACB＝90°.
∵OD∥BC，∴∠AEO＝∠ACB＝90°，即OD⊥AC.
∴＝，∴∠AOD＝∠COD，
∴∠CAD＝∠COD＝∠AOD＝20°.
(2)由(1)可知OD⊥AC，
∴AE＝AC＝×8＝4.
设OA＝x，则OD＝x，∴OE＝x－2.
在Rt△OAE中，OE2＋AE2＝OA2，
即(x－2)2＋42＝x2，解得x＝5，
即OA＝5，∴AB＝2OA＝10.
18．解：(1)(1，－2)
(2)在图中标出点M，D，连结MD、MA，∵M(1，－2)，
D(－3，－2)，A(－2，－1)，
∴MD＝1－(－3)＝4，
MA＝＝.
∵4＞，∴MD＞MA，
∴点D(－3，－2)在⊙M外．
19．解：(1)如图，在等边三角形ABC中，作AD⊥BC于点D.∵AB＝AC，
∴BD＝CD＝BC＝30 cm，
∴AD＝＝30 (cm)，
即正三角形一边的高线长为30 cm.
(2)这个托盘不适用于该花盆．理由如下：
如图，作等边三角形ABC的内切圆⊙O，连结OB，
则点O为等边三角形ABC的内心，
易知点O在AD上，点D在⊙O上，∠OBD＝∠ABC＝30°，
∴在Rt△OBD中，OD＝BD＝10 cm，∴⊙O的直径为20 cm.
∵(20)2＝1 200，352＝1 225，
1 200<1 225，∴20＜35，∴这个托盘不适用于该花盆．
20．(1)证明：∵AD＝BD，DE＝AD，
∴∠DAB＝∠DBA，∠DAE＝∠AED，BD＝ED.
又∵AD＝AD，∠ADB＝∠ADE，
∴△ADB≌△ADE，∴∠AED＝∠ABD＝∠BAD.
∵四边形ABCD内接于⊙O，
∴∠BCD＋∠BAD＝180°，
∴∠BCD＋∠AED＝180°，
∴AE∥BC.
(2)解：如图，连结AC，
则∠ACD＝∠ABD，
∵∠AED＝∠ABD，∠AED＝∠DAE，
∴∠ACD＝∠DAE，
又∵∠AED＝∠AEC，
∴△ECA∽△EAD，
∴＝，∴AE2＝DE·CE.
∵CD＝2，DE＝6，
∴CE＝CD＋DE＝8，
∴AE2＝6×8＝48，
∴AE＝4(负值舍去)，
又∵△ADB≌△ADE，
∴AB＝AE＝4.
21．(1)证明：如图，连结OC.∵OA＝OC，
∴∠ACO＝∠CAD.
∵AC平分∠FAD，
∴∠FAC＝∠CAD，
∴∠FAC＝∠ACO，∴AF∥OC.
∵AF⊥l，∴OC⊥l.
又∵OC是⊙O的半径，
∴直线l是⊙O的切线．
(2)解：如图，连结CD.
∵AD是⊙O的直径，∴∠ACD＝90°.
又∵∠ADC＝∠B＝60°，
∴∠CAD＝30°，∴∠FAC＝∠CAD＝30°，
∴FC＝AC.
设FC＝x，则AC＝2x.
∵AC2－FC2＝AF2，
∴(2x)2－x2＝(4 )2，
解得x＝4(负值舍去)，∴CF＝4.
∵AC平分∠FAD，CG⊥AD，CF⊥AF，∴CG＝CF＝4.
∵∠COG＝2∠CAD＝60°，
∴OC＝＝＝.
∵OC⊥l，∴∠OCE＝90°，
∴∠E＝90°－∠COG＝30°，
∴OE＝2OC＝.
∴S阴影＝S△CEO－S扇形COD＝××4－＝－.
22．解：(1)60
(2)如图，过点O作OH⊥PA，垂足为H，
∵OH⊥PA，∴PH＝PA＝16米，∠POH＝∠POA，
∵摩天轮的直径为40米，∴PO＝20米．
在Rt△POH中，
cos ∠1＝＝＝，
∵cos 37°≈，∴∠1≈37°，
∴∠POH≈90°－37°＝53°，
∴∠POA≈53°×2＝106°，
则弧AP的长约为×π×20＝π(米)，
∴弧AP的长约为米．
(3)t的值约为2.1或11.9.
23．(1)＞
(2)证明：如图①，设PM交⊙O于点C，连结QC，
∵∠PAQ＝∠PCQ，∠PCQ＝
∠PMQ＋∠MQC>∠PMQ，
∴∠PAQ>∠PMQ，
即当运动员跑动到切点A处时，射门张角最大．
(3)解：①如图②所示．
②如图②，根据题意，可知当⊙M与y轴相切于点A时，∠PAQ最大．连结AM，PM，
∵点P(2，0)，点Q(6，0)，∴OP＝2，OQ＝6，∴PQ＝4.
∵MN⊥PQ，∴PN＝NQ＝2，
易知∠AON＝∠ONM＝∠OAM＝90°，∴四边形AONM是矩形，
∴AM＝ON＝2＋2＝4＝PM，
由勾股定理得MN＝＝
2，∴OA＝2，∴点A的坐标为(0，2)．
一、选择题(每题3分，共30分)
1. 若⊙O的半径为4 cm，点A到圆心O的距离为3 cm，则点A与⊙O的位置关系是( )
A．点A在⊙O内 B．点A在⊙O上
C．点A在⊙O外 D．不能确定
2．如图，在⊙O中，＝，∠1＝45°，则∠2等于( )
A．60° B．30° C．45° D．40°
INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\26春9HSJ-9.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\26春9HSJ-9.tif" \* MERGEFORMATINET INCLUDEPICTURE "26春9HSJ-9.tif" \* MERGEFORMAT INCLUDEPICTURE "26春9HSJ-9.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\九SJ-4.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\九SJ-4.tif" \* MERGEFORMATINET INCLUDEPICTURE "九SJ-4.tif" \* MERGEFORMAT INCLUDEPICTURE "九SJ-4.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\9数HSJ-12.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\9数HSJ-12.tif" \* MERGEFORMATINET INCLUDEPICTURE "9数HSJ-12.tif" \* MERGEFORMAT INCLUDEPICTURE "9数HSJ-12.tif" \* MERGEFORMAT
(第2题) (第3题) (第4题)
3．[教材P56习题T9变式]如图，PA，PB分别切⊙O于A，B两点，如果∠P＝60°，PA＝2，那么△PAB的周长为( )
A．2 B．6 C．8 D．4
4．如图，在墙壁中埋着一根半径未知的圆柱形木材，现用锯子去锯这根木材，锯口深CD＝4，锯道AB＝16，则该木材的半径是( )
A．20 B．12 C．10 D．8
5．下列说法正确的是( )
A．三点确定一个圆
B．经过圆心的直线是圆的对称轴
C．和半径垂直的直线是圆的切线
D．三角形的内心到三角形三个顶点的距离相等
6．如图，在Rt△ABC中，∠ACB＝90°，∠ABC＝40°，AB＝6，斜边AB是半圆O的直径，点D是半圆上的一个动点，连结CD与AB交于点E，当BE＝BC时，弧BD的长为( )
A.π B.π C.π D.π
INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\9数HSJ-14.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\9数HSJ-14.tif" \* MERGEFORMATINET INCLUDEPICTURE "9数HSJ-14.tif" \* MERGEFORMAT INCLUDEPICTURE "9数HSJ-14.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\9数HSJ-15.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\9数HSJ-15.tif" \* MERGEFORMATINET INCLUDEPICTURE "9数HSJ-15.tif" \* MERGEFORMAT INCLUDEPICTURE "9数HSJ-15.tif" \* MERGEFORMAT
(第6题) (第7题)
7．如图，AB是⊙O的切线，B为切点，连结AO交⊙O于点C，延长AO交⊙O于点D，连结BD，若∠A＝2∠D，且AB＝4，则AC的长是( )
A．1 B. C．4－2 D．4 －4
8. 如图，点A、B、C、D都在⊙O上，∠C＝120°，AB＝AD＝8，则点O到BD的距离是( )
A. B. C．3 D．4
INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\9数HSJ-16.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\9数HSJ-16.tif" \* MERGEFORMATINET INCLUDEPICTURE "9数HSJ-16.tif" \* MERGEFORMAT INCLUDEPICTURE "9数HSJ-16.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\26春9HSJ-10.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\26春9HSJ-10.tif" \* MERGEFORMATINET INCLUDEPICTURE "26春9HSJ-10.tif" \* MERGEFORMAT INCLUDEPICTURE "26春9HSJ-10.tif" \* MERGEFORMAT
(第8题) (第9题)
9．如图，正八边形ABCDEFGH内接于⊙O，连结AE，HE，若S△AEH＝4，则⊙O的半径为( )
A．2 B．2 C．2 D．4
10. 如图，在菱形ABCD中，AB＝2，∠B＝120°，点O是对角线AC的中点，以点O为圆心，OA长为半径作圆心角为60°的扇形OEF，点D在扇形OEF内，则图中阴影部分的面积为( )
A.－ B．π－
C.－ D．无法确定
二、填空题(每题3分，共15分)
11. 一个圆锥的母线长为6，底面圆的半径为4，那么这个圆锥的侧面积是________. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
12．如图，AB是⊙O的直径，C，D是⊙O上的两点，连结BC，CD，AD，BD，若∠ADC＝25°，BC平分∠ABD，则∠BAD的度数为________. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\9HSJ+1.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\9HSJ+1.tif" \* MERGEFORMATINET INCLUDEPICTURE "9HSJ+1.tif" \* MERGEFORMAT INCLUDEPICTURE "9HSJ+1.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\一改-1.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\一改-1.tif" \* MERGEFORMATINET INCLUDEPICTURE "一改-1.tif" \* MERGEFORMAT INCLUDEPICTURE "一改-1.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\9数HSJ-23.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\9数HSJ-23.tif" \* MERGEFORMATINET INCLUDEPICTURE "9数HSJ-23.tif" \* MERGEFORMAT INCLUDEPICTURE "9数HSJ-23.tif" \* MERGEFORMAT
(第12题) (第13题) (第14题)
13. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\情境题门洞改造.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\情境题门洞改造.tif" \* MERGEFORMATINET INCLUDEPICTURE "情境题门洞改造.tif" \* MERGEFORMAT INCLUDEPICTURE "情境题门洞改造.tif" \* MERGEFORMAT 如图，墙上有一个矩形门洞，宽1.5 m，高2 m，现要将其改造成圆弧形门洞，则改造后圆弧形门洞的最大高度是__________m. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
14. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\新考法新函数思想.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\新考法新函数思想.tif" \* MERGEFORMATINET INCLUDEPICTURE "新考法新函数思想.tif" \* MERGEFORMAT INCLUDEPICTURE "新考法新函数思想.tif" \* MERGEFORMAT 如图，在平面直角坐标系xOy中，若动点P在抛物线y＝ax2上，⊙P恒过点F(0，2)，且与直线y＝－2始终保持相切，则a＝________. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
15．如图，AB＝3，点C是平面内一动点，且BC＝2，连结AC，将AC绕点A逆时针旋转90°，得到AD，连结BD，则BD的最小值为 ________，最大值为 ________． INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
三、解答题(本大题共8个小题，满分75分)
16. (8分)如图，在⊙O中，弦AB与CD相交于点E，AB＝CD.求证： INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
(1)＝；
(2)AE＝CE.
INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\27-15.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\27-15.tif" \* MERGEFORMATINET INCLUDEPICTURE "27-15.tif" \* MERGEFORMAT INCLUDEPICTURE "27-15.tif" \* MERGEFORMAT
17．(9分)如图，AB是半圆O的直径，C，D是半圆O上两点，连结AC，BC，OD，AD，OD∥BC，OD与AC交于点E. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
(1)若∠D＝70°，求∠CAD的度数；
(2)若AC＝8，DE＝2，求AB的长．
INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\27-17.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\27-17.tif" \* MERGEFORMATINET INCLUDEPICTURE "27-17.tif" \* MERGEFORMAT INCLUDEPICTURE "27-17.tif" \* MERGEFORMAT
18．(9分)如图，方格纸上每个小正方形的边长均为1个单位长度，点O，A，B，C均在格点(两条网格线的交点叫格点)上，以点O为原点建立平面直角坐标系. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
(1)过A，B，C三点的圆的圆心M的坐标为________；
(2)请通过计算判断点D(－3，－2)与⊙M的位置关系．
INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\9数HSJ-25.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\9数HSJ-25.tif" \* MERGEFORMATINET INCLUDEPICTURE "9数HSJ-25.tif" \* MERGEFORMAT INCLUDEPICTURE "9数HSJ-25.tif" \* MERGEFORMAT
19.(9分)小陈同学从市场上购买了如图①所示的花盆，花盆底部的横截面是直径为35 cm的圆，他家中有一个托盘，托盘底部的横截面是边长为60 cm的正三角形，如图②所示．
(1)求图②中正三角形一边的高线长；
(2)这个托盘是否适用于该花盆？请通过计算说明理由. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
20．(9分)如图，四边形ABCD内接于⊙O，连结BD，AD＝BD，延长CD至点E，使得DE＝AD，连结AE，∠ADB＝∠ADE.
(1)求证：AE∥BC；
(2)若CD＝2，DE＝6，求AB的长. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\9数HSJ-27.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\9数HSJ-27.tif" \* MERGEFORMATINET INCLUDEPICTURE "9数HSJ-27.tif" \* MERGEFORMAT INCLUDEPICTURE "9数HSJ-27.tif" \* MERGEFORMAT
21．(10分)如图，△ABC内接于⊙O，∠B＝60°.连结AO并延长，与⊙O交于点D，直线l过点C，AF⊥l，垂足为F，AC平分∠FAD，CG⊥AD，垂足为G，直线l交AD的延长线于点E.
(1)求证：直线l是⊙O的切线；
(2)若AF＝4 ，求图中阴影部分的面积. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
22. (10分) INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\情境题摩天轮.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\情境题摩天轮.tif" \* MERGEFORMATINET INCLUDEPICTURE "情境题摩天轮.tif" \* MERGEFORMAT INCLUDEPICTURE "情境题摩天轮.tif" \* MERGEFORMAT 如图①为某游乐场的摩天轮的简化示意图，将摩天轮看作直径为40米的⊙O，摩天轮上的12个轿厢看作均匀分布在⊙O上的点，乘客从点A位置上下摩天轮，摩天轮按顺时针方向匀速转动，转一周用12分钟，线段MN表示上下摩天轮的平台，OD，OE表示支架，分别过图中两个轿厢B，C的位置，OD＝OE＝24米.
(1)∠BOC＝__________°；
(2)如图②，当乘客到达点P处时，与点A之间恰好相距32米，求轿厢转过的弧AP的长度；
(3)设乘客进入轿厢t分钟时所处位置为点Q，连结DQ，若DQ与⊙O相切，直接写出此时t的值. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\26春9HSJ-14.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\26春9HSJ-14.tif" \* MERGEFORMATINET INCLUDEPICTURE "26春9HSJ-14.tif" \* MERGEFORMAT INCLUDEPICTURE "26春9HSJ-14.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\26春9HSJ-15.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\26春9HSJ-15.tif" \* MERGEFORMATINET INCLUDEPICTURE "26春9HSJ-15.tif" \* MERGEFORMAT INCLUDEPICTURE "26春9HSJ-15.tif" \* MERGEFORMAT
23. (11分) INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\新视角探究操作题.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\新视角探究操作题.tif" \* MERGEFORMATINET INCLUDEPICTURE "新视角探究操作题.tif" \* MERGEFORMAT INCLUDEPICTURE "新视角探究操作题.tif" \* MERGEFORMAT 如图，某足球场上，球员在练习射门．线段PQ表示球门，∠PAQ、∠PBQ为射门张角．理论上当射门张角越大时，进球概率越大. INCLUDEPICTURE "C:\\Users\\16979\\Desktop\\得分.tif" \* MERGEFORMAT INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\26春 点拨训练 9 数学 HS（\\得分.tif" \* MERGEFORMATINET INCLUDEPICTURE "得分.tif" \* MERGEFORMAT INCLUDEPICTURE "得分.tif" \* MERGEFORMAT
(1)如图①，∠PAQ__________∠PBQ；(填“>”“<”或“＝”)
(2)若球员沿着直线l带球跑动，如图②，以线段PQ为弦作⊙O，恰与直线l相切于点A.若点M是l上一个异于点A的动点，求证：当运动员跑动到切点A处时，射门张角最大，即∠PAQ>∠PMQ.
(3)如图③，点P(2，0)，点Q(6，0)，点A为y轴正半轴上的一个动点．
①请用尺规作图在y轴上找出点A的位置，使∠PAQ最大(保留作图痕迹，不写作法)；
②当∠PAQ最大时，请求出点A的坐标．
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答案
1．A 2.C 3.B 4.C 5.B 6.B 7.D 8.A 9.B 10.A 11.24π
12. 40° 13. 2.25 14. 15.4；8
16．证明：(1)∵AB＝CD，
∴＝，
∴＋＝＋，∴＝.
(2)连结AD，BC.∵＝，
∴AD＝BC.
∵＝，∴∠ADE＝∠CBE，
又∵∠AED＝∠CEB，
∴△ADE≌△CBE，∴AE＝CE.
17．解：(1)连结OC.∵OA＝OD，
∴∠OAD＝∠D＝70°，
∴∠AOD＝180°－∠OAD－∠D＝40°.
∵AB是半圆O的直径，
∴∠ACB＝90°.
∵OD∥BC，∴∠AEO＝∠ACB＝90°，即OD⊥AC.
∴＝，∴∠AOD＝∠COD，
∴∠CAD＝∠COD＝∠AOD＝20°.
(2)由(1)可知OD⊥AC，
∴AE＝AC＝×8＝4.
设OA＝x，则OD＝x，∴OE＝x－2.
在Rt△OAE中，OE2＋AE2＝OA2，
即(x－2)2＋42＝x2，解得x＝5，
即OA＝5，∴AB＝2OA＝10.
18．解：(1)(1，－2)
(2)在图中标出点M，D，连结MD、MA，∵M(1，－2)，
D(－3，－2)，A(－2，－1)，
∴MD＝1－(－3)＝4，
MA＝＝.
∵4＞，∴MD＞MA，
∴点D(－3，－2)在⊙M外．
19．解：(1)如图，在等边三角形ABC中，作AD⊥BC于点D.∵AB＝AC，
∴BD＝CD＝BC＝30 cm，
∴AD＝＝30 (cm)，
即正三角形一边的高线长为30 cm.
(2)这个托盘不适用于该花盆．理由如下：
如图，作等边三角形ABC的内切圆⊙O，连结OB，
则点O为等边三角形ABC的内心，
易知点O在AD上，点D在⊙O上，∠OBD＝∠ABC＝30°，
∴在Rt△OBD中，OD＝BD＝10 cm，∴⊙O的直径为20 cm.
∵(20)2＝1 200，352＝1 225，
1 200<1 225，∴20＜35，∴这个托盘不适用于该花盆．
20．(1)证明：∵AD＝BD，DE＝AD，
∴∠DAB＝∠DBA，∠DAE＝∠AED，BD＝ED.
又∵AD＝AD，∠ADB＝∠ADE，
∴△ADB≌△ADE，∴∠AED＝∠ABD＝∠BAD.
∵四边形ABCD内接于⊙O，
∴∠BCD＋∠BAD＝180°，
∴∠BCD＋∠AED＝180°，
∴AE∥BC.
(2)解：如图，连结AC，
则∠ACD＝∠ABD，
∵∠AED＝∠ABD，∠AED＝∠DAE，
∴∠ACD＝∠DAE，
又∵∠AED＝∠AEC，
∴△ECA∽△EAD，
∴＝，∴AE2＝DE·CE.
∵CD＝2，DE＝6，
∴CE＝CD＋DE＝8，
∴AE2＝6×8＝48，
∴AE＝4(负值舍去)，
又∵△ADB≌△ADE，
∴AB＝AE＝4.
21．(1)证明：如图，连结OC.∵OA＝OC，
∴∠ACO＝∠CAD.
∵AC平分∠FAD，
∴∠FAC＝∠CAD，
∴∠FAC＝∠ACO，∴AF∥OC.
∵AF⊥l，∴OC⊥l.
又∵OC是⊙O的半径，
∴直线l是⊙O的切线．
(2)解：如图，连结CD.
∵AD是⊙O的直径，∴∠ACD＝90°.
又∵∠ADC＝∠B＝60°，
∴∠CAD＝30°，∴∠FAC＝∠CAD＝30°，
∴FC＝AC.
设FC＝x，则AC＝2x.
∵AC2－FC2＝AF2，
∴(2x)2－x2＝(4 )2，
解得x＝4(负值舍去)，∴CF＝4.
∵AC平分∠FAD，CG⊥AD，CF⊥AF，∴CG＝CF＝4.
∵∠COG＝2∠CAD＝60°，
∴OC＝＝＝.
∵OC⊥l，∴∠OCE＝90°，
∴∠E＝90°－∠COG＝30°，
∴OE＝2OC＝.
∴S阴影＝S△CEO－S扇形COD＝××4－＝－.
22．解：(1)60
(2)如图，过点O作OH⊥PA，垂足为H，
∵OH⊥PA，∴PH＝PA＝16米，∠POH＝∠POA，
∵摩天轮的直径为40米，∴PO＝20米．
在Rt△POH中，
cos ∠1＝＝＝，
∵cos 37°≈，∴∠1≈37°，
∴∠POH≈90°－37°＝53°，
∴∠POA≈53°×2＝106°，
则弧AP的长约为×π×20＝π(米)，
∴弧AP的长约为米．
(3)t的值约为2.1或11.9.
23．(1)＞
(2)证明：如图①，设PM交⊙O于点C，连结QC，
∵∠PAQ＝∠PCQ，∠PCQ＝
∠PMQ＋∠MQC>∠PMQ，
∴∠PAQ>∠PMQ，
即当运动员跑动到切点A处时，射门张角最大．
(3)解：①如图②所示．
②如图②，根据题意，可知当⊙M与y轴相切于点A时，∠PAQ最大．连结AM，PM，
∵点P(2，0)，点Q(6，0)，∴OP＝2，OQ＝6，∴PQ＝4.
∵MN⊥PQ，∴PN＝NQ＝2，
易知∠AON＝∠ONM＝∠OAM＝90°，∴四边形AONM是矩形，
∴AM＝ON＝2＋2＝4＝PM，
由勾股定理得MN＝＝
2，∴OA＝2，∴点A的坐标为(0，2)．