八年级数学试卷 第1页（共92013/2014学年度第一学期期末质量检测八年级数学试卷（时间：100分钟；满分：120分）一、选择题（本大题共8小题，每小题有且只有一个答案正确，请把你认为正确的答案前的字母填入下表相应的空格内，每小题3分，共24分） 1．2的算术平方根是 ······························································································ （ ▲ ） A B ．2 C D ．±2 2．2013年12月2日，“嫦娥三号”从西昌卫星发射中心发射升空，并于12月14日在月球上成功实施软着陆．月球距离地球平均为384401000米，用四舍五入法取近似值，精确到1000000米，并用科学计数法表示，其结果是 ········································ （ ▲ ） A ．3.84×107米 B ．3.8×107米 C ．3.84×108米 D ．3.8×108米3．在实数：213. ，π−227中，无理数的个数有 ······································· （ ▲ ）A ．1个B ．2个C ．3个D ．4个 4．在平面直角坐标系中，点P （3，−5）在 ························································· （ ▲ ） A ．第一象限 B ．第二象限 C ．第三象限 D ．第四象限 5．如图是一个风筝设计图，其主体部分（四边形ABCD ）关于BD 所在的直线对称，AC 与BD 相交于点O ，且AB ≠AD ，则下列判断不正确的是 ······························ （ ▲ ） A ．△ABD ≌△CBD B ．△ABC 是等边三角形 C ．△AOB ≌△COB D ．△AOD ≌△COD6．一次函数y ＝kx b ，当k ＜0，b ＜0时，它的图象大致为 ··························· （ ▲ ）7．如图，正方形网格中，已有两个小正方形被涂黑，再将图中其余小正方形涂黑一个，使整个被涂黑的图案构成一个轴对称图形，那么涂法共有 ···························· （ ▲ ） A ．3种 B ．4种 C ．5种 D ．6种8．某物流公司的快递车和货车同时从甲地出发，以各自的速度匀速向乙地行驶，快递车到达乙地后卸完物品再另装货物共用34h ，立即按原路以另一速度返回，直至与货车相遇．已知货车的速度为60km/h ，两车之间的距离y （km ）与货车行驶时间x （h ）之间的函数图象如图所示，现有以下4个结论： ①快递车到达乙地时两车相距120km ； ②甲、乙两地之间的距离为300km ；第7题 第5题 A B C Dh第8题八年级数学试卷 第2页（共9页）③快递车从甲地到乙地的速度为100km/h ；④图中点B 的坐标为（334，75）．其中，正确的结论有 ························································································· （ ▲ ） A ．1个 B ．2 C ．3个 D ．4个 二、填空题（本大题共10小题，每小题2分，共20分） 9．点P （2-，3-）到x 轴的距离是＿＿＿＿＿．10．比较大小：7．（填“＞”、“＝”、“＜”）11．已知等腰三角形的一个外角是80°，则它顶角的度数为＿＿＿＿＿．12．若直角三角形的两条直角边的长分别是6和8，则斜边上的中线长为＿＿＿＿＿． 13．如图，在△ABC 中，∠C ＝90°，AD 平分∠CAB ，BC ＝10cm ，BD ＝6cm ，那么D 点到直线AB 的距离是＿＿＿＿＿cm ．14．在平面直角坐标系中，一青蛙从点A （−1，0）处向左跳2个单位长度，再向下跳2个单位长度到点A ′处，则点A ′的坐标为＿＿＿＿＿．15．写出同时具备下列两个条件的一次函数关系式＿＿＿＿＿．（写出一个即可） （1）y 随x 的增大而减小；（2）图像经过点（1，−2）． 16．如图，在△ABC 中，AB 的垂直平分线分别交AB 、BC 于点D 、E ，AC 的垂直平分线分别交AC 、BC 于点F 、G ，若∠BAC ＝100°，则∠EAG ＝＿＿＿＿＿°．17．如图，已知直线y ＝ax b -，则关于x 的方程1ax -＝b 的解x ＝＿＿＿＿＿． 18．如图，C 为线段AB 上一动点（不与点A 、B 重合），在AB 同侧分别作正三角形ACD和正三角形BCE ，AE 与BD 交于点F ，AE 与CD 交于点G ，BD 与CE 交于点H ，连接GH ．以下五个结论：①AE ＝BD ；②GH ∥AB ；③AD ＝DH ；④GE ＝HB ；⑤∠AFD ＝60°，一定成立的有＿＿＿＿＿＿＿．（填序号即可）三、解答题（本大题共9小题，共76分，解答要求写出文字说明，证明过程或计算步骤） 19．（本题满分8分） （1）求x 的值：249x -＝0；（2）计算：0(1)-+20．（本题满分6分之间建一座定点医疗站P 交公路内（如图所示）①使其到两公路的距离相等； ②到张、李两村的距离也相等．请你利用尺规作图确定P 点的位置． （不写作法，保留作图痕迹）A B C D 第13题 A B C D E F G 第16题 第17题 A B C D EFG H 第18题21．（本题满分6分）如图，一木杆在离地某处断裂，木杆顶部落在离木杆底部8米处，已知木杆原长16米，求木杆断裂处离地面多少米？22．（本题满分6分）在平面直角坐标系中，已知A（−1，5）、B（4，2）、C（−1，0）三点．（1）点A关于原点O的对称点A′的坐标为＿＿＿＿＿，点B关于x轴的对称点B′的坐标为＿＿＿＿＿，点C关于y轴的对称点C′的坐标为＿＿＿＿＿；（2）求以（1）中的点A′、B′、C′为顶点的△A′B′C′的面积．23．（本题满分6分）如图，四边形ABCD是梯形，AD∥BC，∠A＝90°，BD＝CB，CE⊥BD，垂足为E．（1）求证：△ABD≌△ECB；（2）若∠DBC＝50°，求∠DCE的度数．24．（本题满分10分）如图，点E是∠AOB的平分线上一点，EC⊥OA，ED⊥OB，垂足分别是C、D．求证：（1）∠EDC＝∠ECD；（2）OC＝OD；（3）OE是线段CD的垂直平分线．AB CDEABDEO八年级数学试卷第3页（共9页）第4页（共9页）25．（本题满分10分）阅读下列一段文字，然后回答下列问题．已知平面内两点M （1x ，1y ）、N （2x ，2y ），则这两点间的距离可用下列公式计算：MN例如：已知P （3，1）、Q （1，−2），则这两点间的距离PQ ．特别地，如果两点M （1x ，1y ）、N （2x ，2y ）所在的直线与坐标轴重合或平行于坐标轴或垂直于坐标轴，那么这两点间的距离公式可简化为MN ＝12x x -或12y y -． （1）已知A （1，2）、B （−2，−3），试求A 、B 两点间的距离；（2）已知A 、B 在平行于x 轴的同一条直线上，点A 的横坐标为5，点B 的横坐标为−1，试求A 、B 两点间的距离；（3）已知△ABC 的顶点坐标分别为A （0，4）、B （−1，2）、C （4，2），你能判定△ABC的形状吗？请说明理由． 26．（本题满分12分）小华和爸爸上山游玩，爸爸乘电缆车，小华步行，两人相约在山顶的缆车终点会合．已知小华行走到缆车终点的路程是爸爸乘缆车到山顶的线路长的2倍，爸爸在小华出发后50min 才乘上电缆车，电缆车的平均速度为180m/min ．设小华出发x （min ）行走的路程为y （m ），图中的折线表示小华在整个行走过程中y （m ）与x （min ）之间的函数关系．（1）小华行走的总路程是＿＿＿＿＿m ，他途中休息了＿＿＿＿＿min ； （2）当50≤x ≤80时，求y 与x 的函数关系式；（3）当爸爸到达缆车终点时，小华离缆车终点的路程是多少？八年级数学试卷 第5页（共9页）27．（本题满分12分）已知△ABC 为等边三角形，点D 为直线BC 上的一动点（点D 不与B 、C 重合），以AD 为边作等边△ADE （顶点A 、D 、E 按逆时针方向排列），连接CE ． （1）如图1，当点D 在边BC 上时，求证：①BD ＝CE ，②AC ＝CE ＋CD ；（2）如图2，当点D 在边BC 的延长线上且其他条件不变时，结论AC ＝CE ＋CD 是否成立？若不成立，请写出AC 、CE 、CD 之间存在的数量关系，并说明理由； （3）如图3，当点D 在边BC 的反向延长线上且其他条件不变时，补全图形，并直接写出AC 、CE 、CD 之间存在的数量关系．A BCDEA B C D E A BCD八年级数学参考答案及评分标准（阅卷前请认真校对，以防答案有误！）一、选择题（每小题3分，共24分）二、填空题（每小题2分，共20分）八年级数学试卷第6页（共9页）八年级数学试卷 第7页（共9页）9．3． 10．＜． 11．100°． 12．5．13．4． 14．（−3，−2）．15．答案不唯一，如y ＝1x --等． 16．20． 17．4．18．①②④⑤．三、解答题（共76分）19．（1）24x ＝9， ················································································································ 1分2x ＝94，·············································································································· 2分 x ＝±32． ········································································································· 4分 （2）原式＝1＋2＋2 ······································································································· 3分＝5． ··············································································································· 4分说明：第（1）题答案写成x ＝32扣1分；第（2）题0(1)-的计算分别给1分．20．作出线段垂直平分线， ································································································· 3分作出角平分线． ············································································································· 6分 21．设木杆断裂处离地面x 米，由题意得 ··········································································· 1分228x +＝2(16)x -． ······································································································ 3分 解得x ＝6 ······················································································································· 5分答：木杆断裂处离地面6米． ······················································································ 6分 22．（1）（1，−5）；（4，−2）；（1，0）． ·············································································· 3分（2）S △A ′B ′C ′＝15(41)2⨯⨯-＝152． ···················································· 6分23．（1）∵AD ∥BC ，∴∠ADB ＝∠EBC ． ∵CE ⊥BD ， ∴∠BEC ＝90°． ∵∠A ＝90°，∴∠A ＝∠BEC ． ··········································································································· 1分 在△ABD 和△ECB 中，A BEC ADB EBC BD CB ∠=∠⎧⎪∠=∠⎨⎪=⎩，········································································································ 2分 ∴△ABD ≌△ECB （AAS ）． ························································································ 3分 （2）∵BD ＝CB ，∠DBC ＝50°，∴∠BDC ＝1(180)2DBC ︒-∠＝1(18050)2︒-︒＝65°． ················································· 4分∴在Rt △CDE 中，∠DCE ＝90°-∠BDC ＝90°-65°＝25°． ····································· 6分 24．（1）∵点E 是∠AOB 的平分线上一点，EC ⊥OA ，ED ⊥OB ，∴ED ＝EC ． ·················································································································· 3分 ∴∠EDC ＝∠ECD ． ····································································································· 4分 （2）∵EC ⊥OA ，ED ⊥OB ， ∴∠EDO ＝∠ECO ＝90°． ···························································································· 5分八年级数学试卷 第8页（共9页）由（1）知∠EDC ＝∠ECD ，∴∠EDO -∠EDC ＝∠ECO -∠ECD ，即∠ODC ＝∠OCD ． ···································· 6分 ∴OC ＝OD ． ················································································································· 7分 （3）∵OC ＝OD ，∠EOC ＝∠EOD ，∴OE ⊥CD ，OE 平分CD ，即OE 是线段CD 的垂直平分线． ································ 10分 25．（1）AB········································································· 3分（2）AB ＝5(1)--＝6． ································································································ 6分 （3）△ABC 是直角三角形． ························································································· 7分 理由：∵AB，BC 5，AC∴AB2＋AC 2＝22+＝25，BC 2＝52＝25．∴AB 2＋AC 2＝BC 2． ······································································································ 9分 ∴△ABC 是直角三角形． ··························································································· 10分 26．（1）3600，20． ············································································································· 2分 （2）当50≤x ≤80时，设y 与x 的函数关系式为y ＝kx b +，根据题意得 ·············· 3分当x ＝50时，y ＝1950；当x ＝80时，y ＝3600． ··················································· 4分 ∴501950803600k b k b +=⎧⎨+=⎩． 解得55800k b =⎧⎨=-⎩． ··········································································································· 6分∴y 与x 的函数关系式为y ＝55800x -． ···································································· 7分 （3）缆车到山顶的路线长为3600÷2＝1800（m ）． ···················································· 8分 缆车到达终点所需时间为1800÷180＝10（min ）． ···················································· 9分 爸爸到达缆车终点时，小华行走的时间为10＋50＝60（min ）． ····························· 10分 把x ＝60代入y ＝55800x -，得y ＝55×60-800＝2500． ···································· 11分 ∴当爸爸到达缆车终点时，小华离缆车终点的路程是3600-2500＝1100（m ） ···· 12分 27．（1）∵△ABC 和△ADE 都是等边三角形，∴AB ＝AC ＝BC ，AD ＝AE ，∠BAC ＝∠DAE ＝60°．∴∠BAC -∠CAD ＝∠DAE -∠CAD ，即∠BAD ＝∠CAE ． ······································ 1分 在△ABD 和△ACE 中，AB AC BAD CAE AD AE =⎧⎪∠=∠⎨⎪=⎩， ∴△ABD ≌△ACE （SAS ）． ························································································· 3分 ∴BD ＝CE ． ·················································································································· 4分 ∵BC ＝BD ＋CD ，AC ＝BC ， ∴AC ＝CE ＋CD ． ········································································································· 5分 （2）AC ＝CE ＋CD 不成立，AC 、CE 、CD 之间存在的数量关系是：AC ＝CE -CD ． ············································ 6分 理由：∵AB ＝AC ＝BC ，AD ＝AE ，∠BAC ＝∠DAE ＝60°． ∴∠BAC ＋∠CAD ＝∠DAE ＋∠CAD ，即∠BAD ＝∠CAE ． ······································ 7分。