徐州市 2014至2015学年度期末抽测 全市统考 八年级

（第19题）

2014~2015学年度第一学期期末抽测

八年级数学参考答案

15．AB AC =(或BD CE =、B C ∠=∠、ADC AEB ∠=∠、BDC CEB ∠=∠) 16．30 17．（1）原式＝13(2)-+- ……………3分 （2）24x = ········································· 6分

＝4-．……………………4分 2x =±． ···································· 8分

18．（1）图略（画对即可，各2分）4分 （2）√． ……………………………………………………6分 19．（1）如图； …………………………………………………6分

（注：图像与坐标轴的4个交点、2条直线，各1分） （2）x ＞2． …………………………………………………8分 20．（1）AD BC ⊥． …………………………………………1分 ∵AD 是边BC 上的中线（已知），且BC =6，∴1

32

BD DC BC ===． ·············· 2分

在△ABD 中，∵222222

43255AD BD AB +=+===， ∴△ABD 是直角三角形（勾股定理的逆定理）． ··················································· 4分 ∴90ADB AD BC ∠=?⊥，即． ················································································ 5分 （2）∵AD 是边BC 上的中线，且AD BC ⊥，∴AD 垂直平分BC ． ···························· 7分

∴AC ＝AB ＝5（线段垂直平分线的性质）． ···························································· 8分

21．（1）在△ABC 和△DCB 中，

∵90AB DC

A D BC C

B =??

∠=∠=???=?

（已知），（已知），（公共边）.………3分 ∴△ABC ≌△DCB （HL ）． ············ 5分

（2）∵△ABC ≌△DCB ．∴ACB DBC ∠=∠，即ECB EBC ∠=∠． ····························· 7分

∴EB = EC （等角对等边）． ···················································································· 8分

22．（1）在△ABC 中，∵∠ABC ＝90°，AB =BC ，∴45A C ∠=∠=?．

又∵D 是AC 的中点，∴1

2

BD AC AD ==，BD AC ⊥，45ABD CBD ∠=∠=?．

在△ADE 和△BDF 中，∵45AE BF

DAE DBF AD BD =??

∠=∠=???=?

（已知），（已证），（已证）. ··································· 2分

∴△ADE ≌△BDF （SAS ）．∴DE =DF ． ······························································· 4分 （2）∵△ADE ≌△BDF ，AD E BD F ∠=∠． ·································································· 5分

∵BD AC ⊥（已证），∴90EDF BDE BDF BDE ADE BDA ∠=∠+∠=∠+∠=∠=?．

在△DEF 中，又∵DE =DF （已证），∴=45DEF ∠?． ·········································· 8分

23．（1）根据题意，4个直角三角形全等，小正方形的边长为(b a -)，大正方形边长为c ．

=4S S S ?+大正方形小正方形． ·

························································································· 2分 2222221

4()222

c ab b a ab b ab a a b =?+-=+-+=+．

··········································· 3分 即222a b c +=． ······································································································· 4分 勾股定理：直角三角形两直角边的平方和等于斜边的平方． ······························· 5分

（2）由图知，4S S ?≥大正方形，即21

42

c ab ≥?，由（1）得：222a b ab +≥． ·············· 6分

由图知，小正方形边长为0时，4S S ?=大正方形，此时，0b a b a -==，即． ······ 7分 ∴222a b ab +≥，当a b =时，等号成立． ····························································· 8分

24．（1）2个变量：室内PM2.5的浓度y （mg/m 3）、时间t （h ）； ····································· 2分 （2）点M 表示启动净化器1小时，室内PM2.5浓度达到正常值25 mg/m 3． ············· 4分 （3）设第1小时内，y 与t 的一次函数表达式为y ＝kt b +．根据题意，得

085125k b k b ?+=??

?+=?

，． 解得6085k b =-??=?，

． ∴6085y t =-+． ············································· 7分 （4）根据题意，净化器每小时可使PM2.5的浓度降低60 mg/m 3． ····························· 8分

故所需时间为：2

(12525)6013

-÷=（h ）．

答：需要2

13

h （或100min ）可使PM2.5的浓度恢复正常． ······························· 10分

25．如图，作点A 关于y 轴的对称点'A (1-，4)，连'BA ，交y 轴于点'C ，连'CA ．

由对称性，知'CA CA =． ······························································································· 1分 △ABC 的周长=''AB CA CB AB CA CB AB BA +

+=++≥+（）（）． 线段AB 的长度为定值，当点C 运动到点'C 时，'BC CA +的值最小（两点间线段最短），

此时'='BC CA A B +．故△ABC 的周长的最小值='AB A B +． ···································· 2分 过点'A 作'A D x ⊥轴，垂足为D (1-，0) ．过点A 作AF x ⊥轴，垂足为F (1，0) ． 在Rt △AFB 中，

AB 在Rt △'A DB 中，'A △ABC 设'A B (1)420k b k b -?+=??

?+=?，． 当x =0时，83y =

，即3

∴当点C 运动到'C （0，8

3）时，△ABC 的周长取得最小值 ······················· 8分