新高考數(shù)學(xué)一輪復(fù)習(xí)精講精練7.4 空間距離（提升版）（原卷版）

7.4空間距離（精講）（提升版）思維導(dǎo)圖思維導(dǎo)圖考點(diǎn)呈現(xiàn)考點(diǎn)呈現(xiàn)例題剖析例題剖析考點(diǎn)一點(diǎn)線距【例1】（2022·浙江紹興）如圖，在正三棱柱SKIPIF1<0中，若SKIPIF1<0，則C到直線SKIPIF1<0的距離為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【一隅三反】1．（2022·湖南益陽(yáng)）在棱長(zhǎng)為1的正方體SKIPIF1<0中，SKIPIF1<0為SKIPIF1<0的中點(diǎn)，則點(diǎn)SKIPIF1<0到直線SKIPIF1<0的距離為（

）A．SKIPIF1<0 B．1 C．SKIPIF1<0 D．SKIPIF1<02．（2022·山東）點(diǎn)SKIPIF1<0是直線SKIPIF1<0上一點(diǎn)，SKIPIF1<0是直線SKIPIF1<0的一個(gè)方向向量，則點(diǎn)SKIPIF1<0到直線SKIPIF1<0的距離是______．3．（2022云南）如圖，已知三棱柱SKIPIF1<0的棱長(zhǎng)均為2，SKIPIF1<0，SKIPIF1<0．(1)證明：平面SKIPIF1<0平面ABC；(2)設(shè)M為側(cè)棱SKIPIF1<0上的點(diǎn)，若平面SKIPIF1<0與平面ABC夾角的余弦值為SKIPIF1<0，求點(diǎn)M到直線SKIPIF1<0距離．考點(diǎn)二點(diǎn)面距【例2-1】（2022·哈爾濱）在長(zhǎng)方體SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，則點(diǎn)SKIPIF1<0到平面SKIPIF1<0的距離等于_____.【例2-2】（2022·河北廊坊）如圖所示，在長(zhǎng)方體SKIPIF1<0中，SKIPIF1<0，點(diǎn)E是棱SKIPIF1<0的中點(diǎn)，則點(diǎn)E到平面SKIPIF1<0的距離為（

）A．1 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【一隅三反】1.（2022·江蘇常州）已知正方體SKIPIF1<0的棱長(zhǎng)為2，SKIPIF1<0，SKIPIF1<0分別為上底面SKIPIF1<0和側(cè)面SKIPIF1<0的中心，則點(diǎn)SKIPIF1<0到平面SKIPIF1<0的距離為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022·福建省福州第一中學(xué)高一期末）將邊長(zhǎng)為2的正方形SKIPIF1<0沿對(duì)角線SKIPIF1<0折起，使得平面SKIPIF1<0⊥平面SKIPIF1<0，則點(diǎn)SKIPIF1<0到平面SKIPIF1<0的距離等于（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022·內(nèi)蒙古）如圖，在長(zhǎng)方體SKIPIF1<0中，四邊形SKIPIF1<0是邊長(zhǎng)為2a的正方形，AD＝2AB.(1)若長(zhǎng)方體的表面積為200，求a的值；(2)若a＝1，求點(diǎn)SKIPIF1<0到平面SKIPIF1<0的距離h.考點(diǎn)三線線距【例3】（2022·全國(guó)·高三專(zhuān)題練習(xí)）在長(zhǎng)方體SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則異面直線SKIPIF1<0與SKIPIF1<0之間的距離是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0【一隅三反】1．（2022·山東）定義：兩條異面直線之間的距離是指其中一條直線上任意一點(diǎn)到另一條直線距離的最小值.在長(zhǎng)方體SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則異面直線SKIPIF1<0與SKIPIF1<0之間的距離是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022·江蘇）長(zhǎng)方體SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0為SKIPIF1<0的中點(diǎn)，則異面直線SKIPIF1<0與SKIPIF1<0之間的距離是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022·全國(guó)·高三專(zhuān)題練習(xí)）定義：兩條異面直線之間的距離是指其中一條直線上任意一點(diǎn)到另一條直線距離的最小值．在棱長(zhǎng)為1的正方體SKIPIF1<0中，直線SKIPIF1<0與SKIPIF1<0之間的距離是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0考點(diǎn)四線面距【例4-1】（2022·湖南）在長(zhǎng)方體SKIPIF1<0中，M、N分別為SKIPIF1<0、AB的中點(diǎn)，AB＝4，則MN與平面SKIPIF1<0的距離為_(kāi)_____．【例4-2】（2022廣西）如圖，已知斜三棱柱SKIPIF1<0在底面SKIPIF1<0上的射影恰為SKIPIF1<0的中點(diǎn)SKIPIF1<0又知SKIPIF1<0.(1)求證:SKIPIF1<0平面SKIPIF1<0;(2)求SKIPIF1<0到平面SKIPIF1<0的距離.【一隅三反】1．（2022·江西?。┤鐖D，已知長(zhǎng)方體ABCD-A1B1C1D1中，AB=2，AA1=AD=1，求：(1)平面ADD1A1與平面BCC1B1的距離.(2)點(diǎn)D1到直線AC的距離.(3)直線AB與面A1DCB1的距離.2．（2022·上海市控江中學(xué)）如圖，在長(zhǎng)方體SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0.(1)求直線SKIPIF1<0與平面SKIPIF1<0所成的角的大小；(2)求直線SKIPIF1<0到平面SKIPIF1<0的距離.3．（2022·北京）如圖，已知正方體SKIPIF1<0的棱長(zhǎng)為2，E、F分別是SKIPIF1<0、SKIPIF1<0的中點(diǎn)．(1)求證：SKIPIF1<0平面SKIPIF1<0；(2)在線段BD上是否存在點(diǎn)H，使得EH⊥平面SKIPIF1<0？若存在，求點(diǎn)H的位置；若不存在，說(shuō)明理由；(3)求EF到平面SKIPIF1<0的距離．考點(diǎn)五面面距【例5-1】（2022·全國(guó)·高三專(zhuān)題練習(xí)）在棱長(zhǎng)為SKIPIF1<0的正方體SKIPIF1<0中，則平面SKIPIF1<0與平面SKIPIF1<0之間的距離為A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0【例5-2】（2022·廣東揭陽(yáng)）如圖在直三棱柱SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，E是SKIPIF1<0上的一點(diǎn)，且SKIPIF1<0，D、F、G分別是SKIPIF1<0、SKIPIF1<0、SKIPIF1<0的中點(diǎn)，SKIPIF1<0與SKIPIF1<0相交于SKIPIF1<0．(1)求證：SKIPIF1<0平面SKIPIF1<0；(2)求平面SKIPIF1<0與平面SKIPIF1<0的距離．【一隅三反】1.（2022·全國(guó)·高三專(zhuān)題練習(xí)）已知正方體SKIPIF1<0的棱長(zhǎng)為a，則平面SKIPIF1<0與平面SKIPIF1<0的距離為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022·福建廈門(mén)）如圖，棱長(zhǎng)為2的正方體ABCD–A1B1C1D1中，E，F(xiàn)分別是棱AA1，CC1的中點(diǎn)，過(guò)E作平面SKIPIF1<0，使得SKIPIF1<0//平面BDF.(1)作出SKIPIF1<0截正方體ABCD-A1B1C1D1所得的截面，寫(xiě)出作圖過(guò)程并說(shuō)明理由；(2)求平面SKIPIF1<0與平面SKIPIF1<0的距離.3．（2022·江西?。┤鐖D，在棱長(zhǎng)為a的正方體ABCD－A1B1C1D1中，E、F分別是AA1與CC1的中點(diǎn).(1)證明：平面EB1D1SKIPIF1<0平面FBD；(2)求平面EB1D1與平面FBD之間的距離.7.4空間距離（精練）（提升版）題組一題組一點(diǎn)線距1.（2022·福建）在空間直角坐標(biāo)系中，點(diǎn)SKIPIF1<0，則SKIPIF1<0到直線SKIPIF1<0的距離為_(kāi)__．2（2022·北京·二模）如圖，已知正方體SKIPIF1<0的棱長(zhǎng)為1，則線段SKIPIF1<0上的動(dòng)點(diǎn)P到直線SKIPIF1<0的距離的最小值為3．（2022·廣東）如圖，在棱長(zhǎng)為4的正方體SKIPIF1<0中，E為BC的中點(diǎn)，點(diǎn)P在線段SKIPIF1<0上，點(diǎn)Р到直線SKIPIF1<0的距離的最小值為_(kāi)______.題組二題組二點(diǎn)面距1．（2022·江蘇）將邊長(zhǎng)為SKIPIF1<0的正方形SKIPIF1<0沿對(duì)角線SKIPIF1<0折成直二面角，則點(diǎn)SKIPIF1<0到平面SKIPIF1<0的距離為_(kāi)__．2．（2022·福建福州）如圖，在正四棱柱SKIPIF1<0中，已知SKIPIF1<0，SKIPIF1<0，E，F(xiàn)分別為SKIPIF1<0，SKIPIF1<0上的點(diǎn)，且SKIPIF1<0．(1)求證：SKIPIF1<0平面ACF：(2)求點(diǎn)B到平面ACF的距離．3．（2022·河北邯鄲）在直三棱柱SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0分別是SKIPIF1<0，SKIPIF1<0的中點(diǎn).(1)求證：SKIPIF1<0平面SKIPIF1<0；(2)若SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，求點(diǎn)SKIPIF1<0到平面SKIPIF1<0的距離.4．（2022·四川成都）在四棱錐P-ABCD中，四邊形ABCD為矩形，平面ABCD⊥平面PAB，點(diǎn)E，F(xiàn)分別在線段CB，AP上，且SKIPIF1<0，SKIPIF1<0．(1)求證：SKIPIF1<0平面PCD；(2)若SKIPIF1<0，SKIPIF1<0，求點(diǎn)D到平面EFP的距離．5．（2022·云南保山）如圖，在四棱錐SKIPIF1<0，四邊形SKIPIF1<0正方形，SKIPIF1<0平面SKIPIF1<0．SKIPIF1<0，SKIPIF1<0，點(diǎn)SKIPIF1<0是SKIPIF1<0的中點(diǎn)．(1)求證：SKIPIF1<0平面SKIPIF1<0；(2)求點(diǎn)SKIPIF1<0到平面SKIPIF1<0的距離．題組三題組三線線距1．（2022·全國(guó)·課時(shí)練習(xí)）如圖，多面體SKIPIF1<0是由長(zhǎng)方體一分為二得到的，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，點(diǎn)D是SKIPIF1<0中點(diǎn)，則異面直線SKIPIF1<0與SKIPIF1<0的距離是______．2．（2022·福建）如圖，在正方體SKIPIF1<0中，AB＝1，M，N分別是棱AB，SKIPIF1<0的中點(diǎn)，E是BD的中點(diǎn)，則異面直線SKIPIF1<0，EN間的距離為_(kāi)_____.3．（2022·浙江）如圖，正四棱錐SKIPIF1<0的棱長(zhǎng)均為2，點(diǎn)E為側(cè)棱PD的中點(diǎn)．若點(diǎn)M，N分別為直線AB，CE上的動(dòng)點(diǎn)，則MN的最小值為_(kāi)_____．4．（2022·湖北）如圖，棱長(zhǎng)為1的正方體ABCD-A1B1C1D1中，N是棱AD的中點(diǎn)，M是棱CC1上的點(diǎn)，且CC1=3CM，則直線BM與B1N之間的距離為_(kāi)___.題組四題組四線面距1．（2022·山東濱州）在棱長(zhǎng)為SKIPIF1<0的正方體SKIPIF1<0中，直線BD到平面SKIPIF1<0的距離為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022·山西）如圖，在正方體SKIPIF1<0中，SKIPIF1<0為SKIPIF1<0的中點(diǎn)．(1)證明：SKIPIF1<0平面AD1E(2)求直線SKIPIF1<0到平面SKIPIF1<0的距離；3．（2022·云南·會(huì)澤縣實(shí)驗(yàn)高級(jí)中學(xué)校）如圖，在梯形ABCD中，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0平面ABCD，且SKIPIF1<0，點(diǎn)F在AD上，且SKIPIF1<0．(1)求點(diǎn)A到平面PCF的距離；(2)求AD到平面PBC的距離．題組五題組五面面距1．（2022·江蘇）已知正方體SKIPIF1<0的棱長(zhǎng)為SKIPIF1<0，則平面SKIPIF1<0與平面SKIPIF1<0的距離為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<02．（2022·云南）如圖，在棱長(zhǎng)為1的正方體ABCD-A1B1C1D1中，E，F(xiàn)分別為棱AA1，BB1的中點(diǎn)，則A1B1到平面D1EF的距離是________．3．（2022·上海）如圖，在棱長(zhǎng)為a的正方體SKIPIF1<0中，E、F分別是SKIPIF1<0、SKIPIF1<0的中點(diǎn)．則點(diǎn)A和點(diǎn)SKIPIF1<0的距離為_(kāi)_____，點(diǎn)SKIPIF1<0到棱BC的距離為_(kāi)_____，點(diǎn)E到平面SKIPIF1<0的距離為_(kāi)_____，SKIPIF1<0到平面AEFD的距離為_(kāi)_____．4．（2022·廣東）在棱長(zhǎng)為SKIPIF1<0的正