高一必修2第一章 立体几何练习题

高一必修2第一章 立体几何练习题 第一章 立体几何 第一节 简单几何体 A组 1(下列命题中，不正确的是______( ?棱长都相等的长方体是正方体 ?有两个相邻侧面为矩形的棱柱为直棱柱 ?有两个侧面与底面垂直的棱柱为直棱柱 ?底面为平行四边形的四棱柱叫平行六面体 2((2009年全国卷?改编)纸制的正方体的六个面根据其方位分别标记为上、下、东、南、西、北，现在沿该正方体的一些棱将正方体剪开，外面朝上展平，得到如图的平面图形，则标“?”的面的方位是________( 3((2009年高考安徽卷)对于四面体ABCD，下列命题正确的是________((写出所有正确命题的编号)( ?相对棱AB与CD所在的直线是异面直线; ?由顶点A作四面体的高，其垂足是?BCD三条高线的交点; ?若分别作?ABC和?ABD的边AB上的高，则这两条高的垂足重合; ?任何三个面的面积之和都大于 第四个面的面积; ?分别作三组相对棱中点的连线，所得的三条线段相交于一点( 4(下列三个命题，其中正确的有________个( ?用一个平面去截棱锥，棱锥底面和截面之间的部分是棱台;?两个底面平行且相似，其余各面都是梯形的多面体是棱台;?有两个面互相平行，其余各面都是等腰梯形的六面体是棱台( 5(下面命题正确的有________个( ?长方形绕一条直线旋转一周所形成的几何体是圆柱 ?过圆锥侧面上一点有无数条母线 ?三棱锥的每个面都可以作为底面 ?圆锥的轴截面(过轴所作的截面)是等腰三角形 6.如图所示，长方体的长、宽、高分别为4 cm,3 cm,5 cm，一只蚂蚁从A到C点沿着面1爬行的最短距离是多少, integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, B组 1((2009年高考安徽卷)对于四面体ABCD，下列命题正确的是________( ?相对棱AB与CD所在的直线是异面直线; ?由顶点A作四面体的高，其垂足是?BCD三条高线的交点; ?若分别作?ABC和?ABD的边AB上的高，则这两条高的垂足重合; ?任何三个面的面积之和都大于第四个面的面积; ?分别作三组相对棱中点的连线，所得的三条线段相交于一点( 2(下面是关于三棱锥的四个命题: ?底面是等边三角形，侧面与底面所成的二面角都相等的三棱锥是正三棱锥( ?底面是等边三角形，侧面都是等腰三角形的三棱锥是正三棱锥( ?底面是等边三角形，侧面的面积都相等的三棱锥是正三棱锥( ?侧棱与底面所成的角都相等，且侧面与底面所成的二面角都相等的三棱锥是正三棱锥( 其中，真命题的编号是______((写出所有真命题的编号) 3.关于如图所示几何体的正确说法为________( ?这是一个六面体 ?这是一个四棱台 ?这是一个四棱柱 ?这是一个四棱柱和三棱柱的组合体 ?这是一个被截去一个三棱柱的四棱柱 4((2009年高考安徽卷)对于四面体ABCD，下列命题正确的是 ________( ?相对棱AB与CD所在的直线是异面直线; ?由顶点A作四面体的高，其垂足是?BCD三条高线的交点; ?若分别作?ABC和?ABD的边AB上的高，则这两条高的垂足重合; ?任何三个面的面积之和都大于第四个面的面积; ?分别作三组相对棱中点的连线，所得的三条线段相交于一点( 5(给出以下命题:?底面是矩形的四棱柱是长方体;?直角三角形绕着它的一边旋转一周形成的几何体叫做圆锥;?四棱锥的四个侧面可以都是直角三角形(其中说法正确的是__________( 6(下列结论正确的是 ?各个面都是三角形的几何体是三棱锥 ?以三角形的一条边所在直线为旋转轴，其余两边旋转形成的曲面所围成的几何体叫圆锥 ?棱锥的侧棱长与底面多边形的边长相等，则该棱锥可能是正六棱锥 ?圆锥的顶点与底面圆周上的任意一点的连线都是母线 7(过半径为2的球O表面上一点A作球O的截面，若OA与该截面所成的角是60?，则该截面的面积是________( 8(如果四棱锥的四条侧棱都相等，就称它为“等腰四棱锥”，四条侧棱称为它的腰，以下四个命题中，假命题是________( ?等腰四棱锥的腰与底面所成的角都相等 ?等腰四棱锥的侧面与底面所成的二面角都相等或互补 ?等腰四棱锥的底面四边形必存在外接圆 ?等腰四棱锥的各顶点必在同一球面上 9((2008年高考江西卷)如图(1)，一个正四棱柱形的密闭容器水平 放置，其底部镶嵌了同底的正四棱锥形实心装饰块，容器内盛有a升 水时，水面恰好经过正四棱锥的顶点P.如果将容器倒置，水面也恰好过点P(图(2)) integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, 有下列四个命题: A(正四棱锥的高等于正四棱柱高的一半 B(将容器侧面水平放置时，水面也恰好过点P C(任意摆放该容器，当水面静止时，水面都恰好经过点P D(若往容器内再注入a升水，则容器恰好能装满( 其中真命题的代号是:______(写出所有真命题的代号)( 10(一个四棱锥和一个三棱锥恰好可以拼接成一个三棱柱，这个四棱锥的底面为正方形，且底面边长与各侧棱长相等，这个三棱锥的底面边长与各侧棱长也都相等(设四棱锥、三棱锥、三棱柱的高分别为h，h，h，求h?h?h的值( 123123 11(一个等腰直角三角形的三个顶点分别在正三棱柱的三条侧棱上(已知正三棱柱的底面边长为2，求该三角形的斜边长( 12((2009年高考辽宁卷改编)如果把地球看成一个球体，求地球上北纬60?纬线长和赤道线长的比值( 3 integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand v alue to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, 第二节 空间图形的基本关系与公理 A组 1(以下四个命题中，正确命题的个数是________( ?不共面的四点中，其中任意三点不共线; ?若点A、B、C、D共面，点A、B、C、E共面，则A、B、C、D、E共面; ?若直线a、b共面，直线a、c共面，则直线b、c共面; ?依次首尾相接的四条线段必共面( 2(给出下列四个命题: ?如果两个平面有三个公共点，那么这两个平面重合; ?两条直线可以确定一个平面; ?若M?α，M?β，α?β,l，则M?l; ?空间中，相交于同一点的三条直线在同一平面内( 其中真命题的个数为________( 3((2009年高考湖南卷改编)平行六面体ABCD,ABCD中，既与AB共面也与CC共面11111的棱的条数为________( 4(正方体ABCD,ABCD中，P、Q、R分别是AB、AD、BC的中点(那么，正方体111111 的过P、Q、R的截面图形是________( 5((原创题)已知直线m、n及平面α，其中m?n，那么平面α内到两条直线m、n距离相等的点的集合可能是:(1)一条直线;(2)一个平面;(3)一个点;(4)空集(其中正确的是________( 6(如图，已知平面α、β，且α?β,l.设梯形ABCD中，AD?BC，且AB?α，CD?β.求证:AB，CD，l共点(相交于一点)( B组 1(有以下三个命题: ?平面外的一条直线与这个平面最多有一个公共点; ?直线l在平面α内，可以用符号“l?α”表示; ?若平面α内的一条直线a与平面β内的一条直线b相交，则α与β相交，其中所有正确命题的序号是______________( 2((2010年黄冈调研)下列命题中正确的是________( ?若?ABC在平面α 外，它的三条边所在的直线分别交α于P、Q、R，则P、Q、R三点共线;?若三条直线a、b、c互相平行且分别交直线l于A、B、C三点，则这四条直线共面;?空间中不共面的五个点一定能确定10个平面( 3(对于空间三条直线，有下列四个条件: ?三条直线两两相交且不共点?三条直线两两平行?三条直线共点 ?有两条直线平行，第三条直线和这两条直线都相交 其中使三条直线共面的充分条件有:________. 4((2008年高考浙江卷改编)对两条不相交的空间直线a与b，必存在平面α，使得________( ?a?α，b?α ?a?α，b?α ?a?α，b?α ?a?α，b?α 5(正方体AC中，E、F分别是线段CD、BC的中点，则直线AB与直线EF的位置关111 系是________( 6((2010年湖南郴州调研)设α，β，γ是三个不重合的平面，l是直线，给出下列四个命题: integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, ?若α?β，l?β，则l?α; ?若l?α，l?β，则α?β; ?若l上有两点到α的距离相等，则l?α; ?若α?β，α?γ，则γ?β. 其中正确命题的序号是________( 7((2009年高考广东卷改编)给定下列四个命题: ?若一个平面内的两条直线与另一个平面平行，那么这两个平面相互平行; ?若一个平面经过另一个平面的垂线，那么这两个平面相互垂直; ?垂直于同一直线的两条直线相互平行; ?若两个平面垂直，那么一个平面内与它们的交线不垂直的直线与另一个平面也不垂直( 其中，为真命题的是________( 8((2009年高考宁夏、海南卷改编)如图所示，正方体ABCD,ABCD的棱长为1，线段1111 2BD上有两个动点E，F，且EF,，则下列结论中错误的是________( 112 ?AC?BE ?EF?平面ABCD ?三棱锥A,BEF的体积为定值 ?异面直线AE，BF所成的角为定值 9.(2008年高考陕西卷改编)如图，α?β，α?β=l，A?α，B?β，A、B到l的距离分别是a和b，AB与α、β所成的角分别是θ和φ，AB在α、β内的射影分别是m和n.若a,b，则θ与φ的大小关系为______，m与n的大小关系为______. 10(如图，已知正方体ABCD,ABCD中，E、F分别为DC、111111BC的中点，AC?BD,P，AC?EF,Q，若AC交平面DBFE于11111 R点，试确定R点的位置( 11(如图，在棱长为1的正方体ABCD,ABCD中，M为AB的1111 中点，N为BB的中点，O为平面BCCB的中心( 111 (1)过O作一直线与AN交于P，与CM交于Q(只写作法，不必证明); (2)求PQ的长( 12((2008年高考四川卷)如图，平面ABEF?平面ABCD，四边形 1ABEF与ABCD都是直角梯形，?BAD,?FAB,90?，BC綊AD，2 1BE綊FA，G、H分别为FA、FD的中点( 2 (1)证明:四边形BCHG是平行四边形; (2)C、D、F、E四点是否共面,为什么, (3)设AB,BE，证明:平面ADE?平面CDE. 5 integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, 第三节 平行关系 A组 1(已知m、n是两条不同直线，α，β是两个不同平面，下列命题中的真命题是_( ?如果m?α，n?β，m?n，那么α?β ?如果m?α，n?β，α?β，那么m?n ?如果m?α，n?β，α?β且m，n共面，那么m?n ?如果m?n，m?α，n?β，那么α?β 2(已知m、n是不同的直线，α、β是不重合的平面，给出下列命题: ?若m?α，则m平行于平面α内的无数条直线; ?若α?β，m?α，n?β，则m?n; ?若m?α，n?β，m?n，则α?β; ?若α?β，m?α，则m?β. 其中，真命题的序号是________((写出所有真命题的序号) 3((2010年苏北四市调研)给出下列关于互不相同的直线m、l、n和平面α、β的四个命题: ?若m?α，l?α,A，点A?m, 则l与m不共面; ?若m、l是异面直线，l?α，m?α，且n?l，n?m，则n?α; ?若l?α，m?β，α?β，则l?m; ?若l?α，m?α，l?m,A，l?β，m?β，则α?β. 其中为真命题的是________( 4((2009年高考福建卷改编)设m，n是平面α内的两条不同直线;l，l是平面β内的两12条相交直线，则α?β的一个充分而不必要条件是________( 5((原创题)直线a?平面α，α内有n条直线交于一点，则这n条直线中与直线a平行的直线有________条( 6(如图，ABCD为直角梯形，?C,?CDA,90?，AD,2BC,2CD，P为平面ABCD外一点，且PB?BD. (1)求证:PA?BD; (2)若PC与CD不垂直，求证:PA?PD; (3)若直线l过点P，且直线l?直线BC，试在直线l上找一点E，使得直线PC?平面EBD. B组 1(已知m，n是两条不同的直线，α，β，γ是三个不同的平面，则下列命题正确的是________( ?若α?γ，α?β，则γ?β ?若m?n，m?α，n?β，则α?β ?若m?n，m?α，则n?α ?若n?α，n?β，则α?β 2(已知m，n是两条不同的直线，α，β是两个不同的平面，有下列4个命题: ?若m?n，n?α，则m?α; ?若m?n，m?α，n?α，则n?α; ?若α?β，m?α，n?β，则m?n; ?若m，n是异面直线，m?α，n?β，m?β，则n?α.其中正确的命题有_( 3(已知m，n是平面α外的两条直线，且m?n，则“m?α”是“n?α”的________条件( 4(设l，l是两条直线，α，β是两个平面，A为一点，下列命题中正确的命题是________( 12 ?若l?α，l?α,A，则l与l必为异面直线 1212 ?若α?β，l?α，则l?β 11 ?l?α，l?β，l?β，l?α，则α?β 1212 ?若l?α，l?l，则l?α或l?α 12122 5((2010年广东深圳模拟)若a不平行于平面α，且a?α，则下列结论成立的是________( ?α内的所有直线与a异面 ?α内与a平行的直线不存在 ?α内存在唯一的直线与a平行 integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, ?α内的直线与a都相交 6(设m、n是异面直线，则(1)一定存在平面α，使m?α且n?α;(2)一定存在平面α，使m?α且n?α;(3)一定存在平面γ，使m、n到γ的距离相等;(4)一定存在无数对平面α与β，使m?α，n?β，且α?β.上述4个命题中正确命题的序号为________( 7(如图，ABCD,ABCD是棱长为a的正方体，M、N分别是下底面的棱AB、BC11111111 a的中点，P是上底面的棱AD上的一点，AP,，过P、M、N的平面交上底面于PQ，Q在3 CD上，则PQ,______. 8(下列四个正方体图形中，A、B为正方体的两个顶点，M、N、P分别为其所在棱的中点，能得出AB?面MNP的图形的序号是________(写出所有符合要求的图形序号)( 9(如图所示，在正方体ABCD,ABCD中，E、F、G、H分别是棱CC、CD、DD、11111111CD的中点，N是BC中点(点M在四边形EFGH上及其内部运动，则M满足条件________时，有MN?平面BBDD. 11 ,2，ABABCD中，AA10(如图，长方体ABCD,11111 ,1，AD,2，E为BC的中点，点M为棱AA的中点( 1 (1)证明:DE?平面AAE; 1 (2)证明:BM?平面AED. 1 11((2010年扬州调研)在正方体ABCD,ABCD中，M，N1111 分别是AB，BC的中点( (1)求证:平面BMN?平面BBDD; 111 (2)若在棱DD上有一点P，使BD?平面PMN，求线段DP与11 PD的比 1 12(如图，四边形ABCD为矩形，BC?平面ABE，F为CE上的点，且BF?平面ACE. (1)求证:AE?BE; (2)设点M为线段AB的中点，点N为线段CE的中点(求 证:MN?平面DAE. 7 integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, 第四节 垂直关系 A组 1((2010年宁波十校联考)设b、c表示两条直线，α，β表示两个平面，则下列命题是真命题的是________( ?若b?α，c?α，则b?c ?若b?α，b?c，则c?α ?若c?α，α?β，则c?β ?若c?α，c?β，则α?β 2((2010年青岛质检)已知直线l?平面α，直线m?平面β，下面有三个命题: 3((2009年高考山东卷改编)已知α、β表示两个不同的平面，m为平面α内的一条直线，则“α?β ”是“m?β ”的________条件( 4((2009年高考浙江卷)如图，在长方形ABCD中，AB,2，BC,1，E为DC的中点，F为线段EC(端点除外)上一动点(现将?AFD沿AF折起，使平面ABD?平面ABC.在平面ABD内过点D作DK?AB，K为垂足(设AK,t，则t的取值范围是________( 5((原创题)已知a、b为两条不同的直线，α、β为两个不同的平面，且a?α，b?β，则下列命题中假命题的有________( ?若a?b，则α?β;?若α?β，则a?b;?若a、b相交，则α、β相交;?若α、β相交，则a，b相交( 解 6((2009年高考山东卷)如图，在直四棱柱ABCD,ABCD中，底面ABCD为等腰梯形，AB?CD，AB,4，1111 BC,CD,2，AA,2，E，E分别是棱AD，AA的中点( 111 (1)设F是棱AB的中点，证明:直线EE?平面FCC; 11(2)证明:平面DAC?平面BBCC. 111 B组 1(设a，b是两条不同的直线，α，β是两个不同的平面，则能得出a?b的是____( ?a?α，b?β，α?β ?a?α，b?β，α?β ?a?α，b?β，α?β ?a?α，b?β，α?β 2(设α，β为不重合的平面，m，n为不重合的直线，则下列命题正确的是________( ?若m?α，n?β，m?n，则α?β ?若n?α，n?β，m?β，则m?α ?若m?α，n?β，m?n，则α?β ?若α?β，α?β,n，m?n，则m?α 3(设m，n是两条不同的直线， α，β是两个不同的平面，则下列命题正确的是( ?m?α，n?β，m?n?α?β ?α?β，m?α，n?β ?m?n ?α?β，m?α，n?β ?m?n ?α?β，α?β,m，n?m?n?β 4(已知两条不同的直线m，n，两个不同的平面α，β，则下列命题中正确的是_( integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, ?若m?α，n?β，α?β，则m?n ?若m?α，n?β，α?β，则m?n ?若m?α，n?β，α?β，则m?n ?若m?α，n?β，α?β，则m?n 5(设a，b，c表示三条直线，α，β表示两个平面，则下列命题的逆命题不成立的是________( ?c?α，若c?β，则α?β ?b?β，c是a在β内的射影，若b?c，则a?b ?b?β，若b?α，则β?α ?b?α，c?α，若c?α，则b?c 6(已知二面角α,l,β的大小为30?，m、n为异面直线，m?平面α，n?平面β，则m、n所成的角为________( 7(如图所示，在斜三棱柱ABC,ABC中，?BAC,90?，BC?AC，则C在底面ABC11111上的射影H必在直线______上( 8((2010年江苏昆山模拟)在矩形ABCD中，AB,3，AD,4，P在AD上运动，设?ABP,θ，将?ABP沿BP折起，使得平面ABP垂直于平面BPDC，AC长最小时θ的值为________( 39(在正四棱锥P,ABCD中，PA,AB，M是BC的中点，G是?PAD的重心，则在平2 面PAD中经过G点且与直线PM垂直的直线有________条( 10(如图，在三棱锥S,ABC中，OA,OB，O为BC中点，SO?平面ABC，E为SC中点，F为AB中点( (1)求证:OE?平面SAB; (2)求证:平面SOF?平面SAB. 11(在长方体ABCD,ABCD中，AA,2AB,2BC，E，F，E分别是棱AA，BB，AB1111111111的中点( (1)求证:CE?平面CEF; 11 (2)求证:平面CEF?平面CEF. 11 12((2010年江苏淮安模拟)如图，已知空间四边形ABCD中，BC,AC，AD,BD，E是AB的中点( 求证:(1)AB?平面CDE; (2)平面CDE?平面ABC; (3)若G为?ADC的重心，试在线段AE上确定一点F，使得GF?平面CDE. 9 integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, 第五节 简单几何体的面积和体积 A组 1((2010年东北四校联考)已知一个长方体的同一顶点处的三条棱长分别为1，3，2，则其外接球的表面积为________( 2((2009年高考上海卷)若等腰直角三角形的直角边长为2，则以一直角边所在的直线为轴旋转一周所成的几何体体积是_________( 3((2010年南京调研)如图，在正三棱柱ABC,ABC中，D为棱AA1111的中点(若截面?BCD是面积为6的直角三角形，则此三棱柱的体积为1 ________( 4(矩形ABCD中，AB,4，BC,3，沿AC将矩形ABCD折成一个直二面角B,AC,D，则四面体ABCD的外接球的体积为________( 5(已知过球面上三点A、B、C的截面到球心的距离等于球半径的一半，且AC,BC,6，AB,4，则球的半径等于________，球的表面积等于________( 6(在长方体ABCD,ABCD中，AB,BC,2，过A、C、B111111 三点的平面截去长方体的一个角后，得到如图所示的几何体ABCD 40,ACD，且这个几何体的体积为. 1113 (1)证明:直线AB?平面CDDC; 111 (2)求棱AA的长; 1 (3)求经过A，C，B，D四点的球的表面积( 11 B组 1((2008年高考湖北卷)用与球心距离为1的平面去截球，所得的截面面积为π，则球的体积为________( 2(在三棱锥A,BCD中，侧棱AB、AC、AD两两垂直，?ABC，?ACD，?ADB的面积 236分别为，，，则该三棱锥的体积为________( 222 3((2010年福建厦门检测)已知一个球与一个正三棱柱的三个侧面和两个底面相切，若这 32π个球的体积是，则这个三棱柱的体积是________( 3 4((2009年高考陕西卷改编)若正方体的棱长为2，则以该正方体各个面的中心为顶点的integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, 凸多面体的体积为________( 5((2009年高考全国卷?)已知OA为球O的半径，过OA的中点M且垂直于OA的平面截球面得到圆M.若圆M的面积为3π，则球O的表面积等于__________( 6((2009年高考江西卷)体积为8的一个正方体，其全面积与球O的表面积相等，则球O的体积等于________( 7(若长方体的三个共顶点的面的面积分别是2，3，6，则长方体的体积是__( 8(在一个锥体中，作平行于底面的截面，若这个截面面积与底面面积之比为1?3，则锥体被截面所分成的两部分的体积之比为________ 9((2010年南通调研)正方体ABCD,ABCD的棱长为23，则四面体A,BCD的外接111111球的体积为________( 10((2009年高考宁夏、海南卷)如图，在三棱锥P,ABC中，?PAB是等边三角形，?PAC,?PBC,90?. (1)证明:AB?PC; (2)若PC,4，且平面PAC?平面PBC，求三棱锥P,ABC的体积( 11(如图，已知AB?平面ACD，DE?平面ACD，?ACD为等边三角形，AD,DE,2AB,2，F为CD的中点( (1)求证:AF?平面CDE; (2)求证:AF?平面BCE; (3)求四棱锥C,ABED的体积( 12((2010年广州质检)如图，AA是圆柱的母线，AB是圆柱底面1 圆的直径，C是底面圆周上异于A、B的任意一点，AA,AB,2. 1 (1)求证:BC?平面AAC; 1 (2)求三棱锥A,ABC的体积的最大值( 1 11 integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed, integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the city's liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changning district, shale gas production capacity reached 277 million cubic metres, built the country's first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for "water and the second land of the seven hills". 236 meters to 2000 meters above sea level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, "old town-the South Bank" Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, "suspicious" pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed,