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2020 年上海市嘉定区中考数学一模试卷
参考答案
2020.1
一、选择题（本大题共6题，每题4分，满分24分）
1.D；2.A；3. D；4.A；5.B；6.C.
二、填空题：（本大题共12题，每题4分，满分48分）
7. ；8. ；9. ；10. ；11. ；12. ；13. ；14. ；15.75；16. （不
要求写出函数的定义域）；17. ；18. .
三、解答题（本大题共7题，满分58分）
19.（本题满分10分）
解：
= ·················································································8分
= .··················································································1+1分
20.（本小题满分10分，第（1）小题6分，第（2）小题4分）
证明：在Rt AOH中，
∵ △ ， ， ，
∴ (m).··································································2分
∴ (m) ·····························································2分
在Rt BOH中，
∵ △ ， ， ，
∴ ·····································································2分
A
O
（2）过点 向直线 作垂线，垂足为 .··························································1分
M B
H
图6-2在Rt ABM中，
△
∵ ， ， ，
∴ ·····································································2分
答： 的正弦值为 ，点 到直线 的距离是1米.·······································1分
21.（本题满分10分）
解：设 ， ，·····································································1分
由题意，得 ， ··································································2分
∵ ，∴ = .·············································································1分
∴ ，即 .·········································································2分
∵ 、 、 、 都是⊙O的半径，∴ .····························1分
∵ ，
∴△AOB≌△COD.·························································································2分
∴ .··································································································1分
22.（本题满分10分）
解：过点 作直线 的垂线，垂足为 （如图7所示）········································1分
由题意，得 ， .··························································1分
∴ .···································································1分
又∵ ，∴ .·····································1分
∴ .····························································································1分
∵ ，∴ （海里）··························································1分在Rt△ACD中， （海里）···························2分
由题意知：以海岛A为圆心，半径长为10海里范围内有暗礁.这里， ，
所以，如果货轮继续向东航行，没有触礁的危险. ·············································2分
A
D E
D B C
图7 图8
23.（本题满分12分，第（1）小题4分，第（2）小题8分）
证明：（1）∵ ∥ ，∴ .·······················································1分
又∵ ，∴△BDE∽△CBE. ·························································1分
∴ .························································································1分
∴ .··················································································1分
（2）∵ ∥ ，∴ .又 ，∴ .·····················1分
又∵ ，∴△ADE∽△ABE. ····················································1分
∴ .·······················································································1分
∵ ∥ ，∴ ，即 .···············································1分
∴ .·······················································································1分
∵ 平分 ，∴ ，又∵ ，∴ .·······1分
∴ .··························································································1分
∴ .·······················································································1分
24．（本题满分12分，每小题4分）解：（1）图像基本正确（开口方向、对称轴、顶点、大致光滑）··························2分
将图9中的抛物线 向下平移2个单位长，可得抛物线 ················2分
备注：如果使用“列表、描点、连线”的方式叙述，需要呈现列表使用的表格.
（2）由题意，得点 的“关联点”为 ·······································1分
由点 在抛物线 上，可得 ， ····························1分
又∵ 在抛物线 上，∴ ····································1分
解得 .将 代入 ，得 ··········································1分
（3）点 的“待定关联点”为 ，·············································1分
∵ 在抛物线 的图像上，∴ .······················1分
∴ , .又∵ ，∴ .···············································1分
当 时， ，故可得 .··············································1分
25．（本题满分14分，第（1）小题4分，第（2）、（3）小题各5分）
证明：（1）∵ ， ，······················1分
∴ .···················································1分
即 .
∴ .···················································································1分
又∵ ，∴ ∽ .··························································1分
（2）如图10-1，∵ ， ，∴ .·················1分
在 中，∵ ， ，∴ .····························1分
又∵ ∽ ，∴ .·····················································1分
∴ ，即 .·································································2分
A
A A
A
P P P
P
C
B C B C B
图10-1
图10-2 C B
图10-3 图10-4（3）∵ ， ， ，∴ .
∴ .································1分
∵ ∽ ，∴ ，∴ .
①如图10-2，当 是等腰三角形，且 时， .
···········································································································1分
②如图10-3，当 是等腰三角形，且 时， , ，
易得 ，∴ ·····················································2分
③如图10-4，当 是等腰三角形，且 时， ， ，
易得 ，∴ .···················································1分
备注：写出 ， 这两个答案之中的一个，即可得到2分；两个
全部写出，得3分.
