第第頁(yè)第08講直線的傾斜角與斜率【題型歸納目錄】題型一：直線的傾斜角與斜率定義題型二：斜率與傾斜角的變化關(guān)系題型三：已知兩點(diǎn)求斜率、已知斜率求參數(shù)題型四：直線與線段相交關(guān)系求斜率范圍題型五：直線平行題型六：直線垂直題型七：直線平行、垂直在幾何問(wèn)題的應(yīng)用【知識(shí)點(diǎn)梳理】知識(shí)點(diǎn)一：直線的傾斜角平面直角坐標(biāo)系中，對(duì)于一條與SKIPIF1<0軸相交的直線，如果把SKIPIF1<0軸繞著交點(diǎn)按逆時(shí)針?lè)较蛐D(zhuǎn)到和直線重合時(shí)所轉(zhuǎn)的最小正角記為SKIPIF1<0，則SKIPIF1<0叫做直線的傾斜角．規(guī)定：當(dāng)直線和SKIPIF1<0軸平行或重合時(shí)，直線傾斜角為SKIPIF1<0，所以，傾斜角的范圍是SKIPIF1<0．知識(shí)點(diǎn)詮釋：1、要清楚定義中含有的三個(gè)條件①直線向上方向；②SKIPIF1<0軸正向；③小于SKIPIF1<0的角．2、從運(yùn)動(dòng)變化觀點(diǎn)來(lái)看，直線的傾斜角是由SKIPIF1<0軸按逆時(shí)針?lè)较蛐D(zhuǎn)到與直線重合時(shí)所成的角．3、傾斜角SKIPIF1<0的范圍是SKIPIF1<0．當(dāng)SKIPIF1<0時(shí)，直線與x軸平行或與x軸重合．4、直線的傾斜角描述了直線的傾斜程度，每一條直線都有唯一的傾斜角和它對(duì)應(yīng)．5、已知直線的傾斜角不能確定直線的位置，但是，直線上的一點(diǎn)和這條直線的傾斜角可以唯一確定直線的位置．知識(shí)點(diǎn)二：直線的斜率1、定義：傾斜角不是SKIPIF1<0的直線，它的傾斜角的正切叫做這條直線的斜率，常用SKIPIF1<0表示，即SKIPIF1<0．知識(shí)點(diǎn)詮釋：(1)當(dāng)直線SKIPIF1<0與x軸平行或重合時(shí)，SKIPIF1<0，SKIPIF1<0；(2)直線SKIPIF1<0與x軸垂直時(shí)，SKIPIF1<0，k不存在．由此可知，一條直線SKIPIF1<0的傾斜角SKIPIF1<0一定存在，但是斜率k不一定存在．2、直線的傾斜角SKIPIF1<0與斜率SKIPIF1<0之間的關(guān)系由斜率的定義可知，當(dāng)SKIPIF1<0在SKIPIF1<0范圍內(nèi)時(shí)，直線的斜率大于零；當(dāng)SKIPIF1<0在SKIPIF1<0范圍內(nèi)時(shí)，直線的斜率小于零；當(dāng)SKIPIF1<0時(shí)，直線的斜率為零；當(dāng)SKIPIF1<0時(shí)，直線的斜率不存在．直線的斜率與直線的傾斜角(SKIPIF1<0除外)為一一對(duì)應(yīng)關(guān)系，且在SKIPIF1<0和SKIPIF1<0范圍內(nèi)分別與傾斜角的變化方向一致，即傾斜角越大則斜率越大，反之亦然．因此若需在SKIPIF1<0或SKIPIF1<0范圍內(nèi)比較傾斜角的大小只需比較斜率的大小即可，反之亦然．知識(shí)點(diǎn)三：斜率公式已知點(diǎn)SKIPIF1<0、SKIPIF1<0，且SKIPIF1<0與SKIPIF1<0軸不垂直，過(guò)兩點(diǎn)SKIPIF1<0、SKIPIF1<0的直線的斜率公式SKIPIF1<0．知識(shí)點(diǎn)詮釋：1、對(duì)于上面的斜率公式要注意下面五點(diǎn)：(1)當(dāng)SKIPIF1<0時(shí)，公式右邊無(wú)意義，直線的斜率不存在，傾斜角SKIPIF1<0，直線與SKIPIF1<0軸垂直；(2)SKIPIF1<0與SKIPIF1<0、SKIPIF1<0的順序無(wú)關(guān)，即SKIPIF1<0，SKIPIF1<0和SKIPIF1<0，SKIPIF1<0在公式中的前后次序可以同時(shí)交換，但分子與分母不能交換；(3)斜率SKIPIF1<0可以不通過(guò)傾斜角而直接由直線上兩點(diǎn)的坐標(biāo)求得；(4)當(dāng)SKIPIF1<0時(shí)，斜率SKIPIF1<0，直線的傾斜角SKIPIF1<0，直線與SKIPIF1<0軸平行或重合；(5)求直線的傾斜角可以由直線上兩點(diǎn)的坐標(biāo)先求斜率而得到．2、斜率公式的用途：由公式可解決下列類型的問(wèn)題：(1)由SKIPIF1<0、SKIPIF1<0點(diǎn)的坐標(biāo)求SKIPIF1<0的值；(2)已知SKIPIF1<0及SKIPIF1<0中的三個(gè)量可求第四個(gè)量；(3)已知SKIPIF1<0及SKIPIF1<0、SKIPIF1<0的橫坐標(biāo)(或縱坐標(biāo))可求SKIPIF1<0；(4)證明三點(diǎn)共線．知識(shí)點(diǎn)四：兩直線平行的條件設(shè)兩條不重合的直線SKIPIF1<0的斜率分別為SKIPIF1<0．若SKIPIF1<0，則SKIPIF1<0與SKIPIF1<0的傾斜角SKIPIF1<0與SKIPIF1<0相等．由SKIPIF1<0，可得SKIPIF1<0，即SKIPIF1<0．因此，若SKIPIF1<0，則SKIPIF1<0．反之，若SKIPIF1<0，則SKIPIF1<0．知識(shí)點(diǎn)詮釋：1、公式SKIPIF1<0成立的前提條件是①兩條直線的斜率存在分別為SKIPIF1<0；②SKIPIF1<0不重合；2、當(dāng)兩條直線的斜率都不存在且不重合時(shí)，SKIPIF1<0的傾斜角都是SKIPIF1<0，則SKIPIF1<0．知識(shí)點(diǎn)五：兩直線垂直的條件設(shè)兩條直線SKIPIF1<0的斜率分別為SKIPIF1<0．若SKIPIF1<0，則SKIPIF1<0．知識(shí)點(diǎn)詮釋：1、公式SKIPIF1<0成立的前提條件是兩條直線的斜率都存在；2、當(dāng)一條垂直直線的斜率不存在，另一條直線的斜率為0時(shí)，兩條直線也垂直．【典例例題】題型一：直線的傾斜角與斜率定義例1．直線SKIPIF1<0的傾斜角為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例2．對(duì)于下列命題：①若SKIPIF1<0是直線l的傾斜角，則SKIPIF1<0；②若直線傾斜角為SKIPIF1<0，則它斜率SKIPIF1<0；③任一直線都有傾斜角，但不一定有斜率；④任一直線都有斜率，但不一定有傾斜角．其中正確命題的個(gè)數(shù)為(

)A．1 B．2 C．3 D．4例3．已知傾斜角為SKIPIF1<0的直線SKIPIF1<0與直線SKIPIF1<0的夾角為SKIPIF1<0，則SKIPIF1<0的值為(

)A．SKIPIF1<0或SKIPIF1<0 B．SKIPIF1<0或SKIPIF1<0 C．SKIPIF1<0或SKIPIF1<0 D．SKIPIF1<0或SKIPIF1<0例4．直線SKIPIF1<0經(jīng)過(guò)SKIPIF1<0，SKIPIF1<0兩點(diǎn)，則直線SKIPIF1<0的傾斜角是(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例5．已知直線經(jīng)過(guò)點(diǎn)SKIPIF1<0，SKIPIF1<0，該直線的傾斜角為(

)．A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例6．直線n的傾斜角為150°，則它的斜率k＝()A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0題型二：斜率與傾斜角的變化關(guān)系例7．已知直線過(guò)SKIPIF1<0，SKIPIF1<0兩點(diǎn)，且傾斜角為SKIPIF1<0，則SKIPIF1<0(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例8．若如圖中的直線SKIPIF1<0的斜率為SKIPIF1<0，則(

)

A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例9．直線SKIPIF1<0的傾斜角的取值范圍是(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例10．若過(guò)點(diǎn)SKIPIF1<0，SKIPIF1<0的直線的傾斜角為銳角，則實(shí)數(shù)SKIPIF1<0的取值范圍為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例11．已知直線SKIPIF1<0的斜率為SKIPIF1<0,則SKIPIF1<0的傾斜角為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例12．若直線SKIPIF1<0的斜率為SKIPIF1<0，且SKIPIF1<0，則直線SKIPIF1<0的傾斜角為(

)A．SKIPIF1<0或SKIPIF1<0 B．SKIPIF1<0或SKIPIF1<0 C．SKIPIF1<0或SKIPIF1<0 D．SKIPIF1<0或SKIPIF1<0題型三：已知兩點(diǎn)求斜率、已知斜率求參數(shù)例13．經(jīng)過(guò)點(diǎn)SKIPIF1<0的直線的斜率為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例14．已知SKIPIF1<0，SKIPIF1<0，則直線SKIPIF1<0的斜率為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．7例15．已知直線斜率等于SKIPIF1<0，則該直線的傾斜角為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例16．若過(guò)兩點(diǎn)SKIPIF1<0，SKIPIF1<0的直線的傾斜角為150°，則SKIPIF1<0的值為(

)A．SKIPIF1<0 B．0 C．SKIPIF1<0 D．3例17．若直線經(jīng)過(guò)兩點(diǎn)SKIPIF1<0，SKIPIF1<0，且其傾斜角為135°，則m的值為(

)A．0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例18．設(shè)SKIPIF1<0為實(shí)數(shù)，已知過(guò)兩點(diǎn)SKIPIF1<0，SKIPIF1<0的直線的斜率為SKIPIF1<0，則SKIPIF1<0的值為(

)A．2 B．3 C．4 D．5題型四：直線與線段相交關(guān)系求斜率范圍例19．已知點(diǎn)SKIPIF1<0，直線SKIPIF1<0過(guò)點(diǎn)SKIPIF1<0且與線段SKIPIF1<0相交，則直線SKIPIF1<0的斜率的取值范圍是(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例20．已知SKIPIF1<0、SKIPIF1<0，若直線SKIPIF1<0經(jīng)過(guò)點(diǎn)SKIPIF1<0，且與線段SKIPIF1<0有交點(diǎn)，則SKIPIF1<0的斜率的取值范圍為(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例21．已知點(diǎn)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，若點(diǎn)SKIPIF1<0是線段SKIPIF1<0上的一點(diǎn)SKIPIF1<0，則直線SKIPIF1<0的斜率的取值范圍是(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例22．已知兩點(diǎn)SKIPIF1<0，過(guò)點(diǎn)SKIPIF1<0的直線與線段SKIPIF1<0有交點(diǎn)，則直線SKIPIF1<0的傾斜角的取值范圍為(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例23．經(jīng)過(guò)點(diǎn)SKIPIF1<0作直線SKIPIF1<0，且直線SKIPIF1<0與連接點(diǎn)SKIPIF1<0，SKIPIF1<0的線段總有公共點(diǎn)，則直線SKIPIF1<0的傾斜角SKIPIF1<0的取值范圍是(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例24．已知SKIPIF1<0、SKIPIF1<0，直線SKIPIF1<0過(guò)定點(diǎn)SKIPIF1<0，且與線段SKIPIF1<0相交，則直線SKIPIF1<0的斜率SKIPIF1<0的取值范圍是(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0或SKIPIF1<0題型五：直線平行例25．已知直線l1：x＋my－2m－2＝0，直線l2：mx＋y－1－m＝0，當(dāng)SKIPIF1<0時(shí)，m＝_________例26．已知直線SKIPIF1<0的傾斜角為SKIPIF1<0，直線SKIPIF1<0經(jīng)過(guò)點(diǎn)SKIPIF1<0，SKIPIF1<0，則直線SKIPIF1<0與SKIPIF1<0的位置關(guān)系是______．例27．直線SKIPIF1<0的傾斜角為SKIPIF1<0，直線SKIPIF1<0過(guò)SKIPIF1<0，SKIPIF1<0，則直線SKIPIF1<0與SKIPIF1<0的位置關(guān)系為______.例28．已知直線SKIPIF1<0：SKIPIF1<0，SKIPIF1<0：SKIPIF1<0.當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0___________.例29．已知兩條直線SKIPIF1<0和SKIPIF1<0互相平行，則正數(shù)a的值為____.例30．已知直線SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0，則實(shí)數(shù)SKIPIF1<0______．題型六：直線垂直例31．直線SKIPIF1<0過(guò)點(diǎn)SKIPIF1<0和點(diǎn)SKIPIF1<0，直線SKIPIF1<0過(guò)點(diǎn)SKIPIF1<0和點(diǎn)SKIPIF1<0，則直線SKIPIF1<0與SKIPIF1<0的位置關(guān)系是______．例32．已知SKIPIF1<0三點(diǎn)，則△ABC為__________三角形.例33．已知兩條直線SKIPIF1<0的斜率是方程SKIPIF1<0的兩個(gè)根，則SKIPIF1<0與SKIPIF1<0的位置關(guān)系是______例34．已知直線SKIPIF1<0與直線SKIPIF1<0垂直，則實(shí)數(shù)a的值為________．例35．已知經(jīng)過(guò)點(diǎn)SKIPIF1<0和點(diǎn)SKIPIF1<0的直線l1與經(jīng)過(guò)點(diǎn)SKIPIF1<0和點(diǎn)SKIPIF1<0的直線SKIPIF1<0互相垂直，則實(shí)數(shù)SKIPIF1<0_____.例36．已知直線SKIPIF1<0經(jīng)過(guò)點(diǎn)SKIPIF1<0，直線SKIPIF1<0經(jīng)過(guò)點(diǎn)SKIPIF1<0，若SKIPIF1<0，則SKIPIF1<0的值為________________.題型七：直線平行、垂直在幾何問(wèn)題的應(yīng)用例37．在平面直角坐標(biāo)系SKIPIF1<0中，設(shè)三角形SKIPIF1<0的頂點(diǎn)分別為SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，點(diǎn)SKIPIF1<0是線段SKIPIF1<0上的一點(diǎn)(異于端點(diǎn))，設(shè)SKIPIF1<0均為非零實(shí)數(shù)，直線SKIPIF1<0分別交SKIPIF1<0于點(diǎn)SKIPIF1<0，若SKIPIF1<0，求證：SKIPIF1<0.

例38．已知SKIPIF1<0，A，B，C，D四點(diǎn)構(gòu)成的四邊形是平行四邊形，求點(diǎn)D的坐標(biāo).例39．在平面直角坐標(biāo)系中，四邊形SKIPIF1<0的頂點(diǎn)按逆時(shí)針順序依次是SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，其中SKIPIF1<0，試判斷四邊形SKIPIF1<0的形狀，并給出證明．例40．已知SKIPIF1<0，SKIPIF1<0，SKIPIF1<0.(1)若SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0可以構(gòu)成平行四邊形，求點(diǎn)SKIPIF1<0的坐標(biāo)；(2)在(1)的條件下，判斷SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0構(gòu)成的平行四邊形是否為菱形.例41．已知四邊形ABCD的頂點(diǎn)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，是否存在點(diǎn)A，使四邊形ABCD為直角梯形？若存在，求出點(diǎn)A的坐標(biāo)；若不存在，請(qǐng)說(shuō)明理由．【過(guò)關(guān)測(cè)試】一、單選題1．若直線SKIPIF1<0與直線SKIPIF1<0垂直，則實(shí)數(shù)SKIPIF1<0(

)A．0 B．1 C．SKIPIF1<0 D．SKIPIF1<02．已知直線l的傾斜角為SKIPIF1<0，直線SKIPIF1<0經(jīng)過(guò)點(diǎn)SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0與l垂直，直線SKIPIF1<0與直線SKIPIF1<0平行，則SKIPIF1<0等于(

)A．SKIPIF1<0 B．SKIPIF1<0 C．0 D．23．已知兩條直線SKIPIF1<0，SKIPIF1<0，則SKIPIF1<0是SKIPIF1<0的(

)A．必要不充分條件 B．充分不必要條件C．充要條件 D．既不充分也不必要條件4．以SKIPIF1<0為頂點(diǎn)的四邊形是(

)A．平行四邊形，但不是矩形B．矩形C．梯形，但不是直角梯形D．直角梯形5．已知坐標(biāo)平面內(nèi)三點(diǎn)SKIPIF1<0，SKIPIF1<0為SKIPIF1<0的邊SKIPIF1<0上一動(dòng)點(diǎn)，則直線SKIPIF1<0斜率SKIPIF1<0的變化范圍是(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<06．已知SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0在線段SKIPIF1<0上，則SKIPIF1<0的最小值為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．若過(guò)點(diǎn)SKIPIF1<0，SKIPIF1<0的直線的傾斜角為銳角，則實(shí)數(shù)SKIPIF1<0的取值范圍為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<08．已知傾斜角為SKIPIF1<0的直線SKIPIF1<0與直線SKIPIF1<0的夾角為SKIPIF1<0，則SKIPIF1<0的值為(

)A．SKIPIF1<0或SKIPIF1<0 B．SKIPIF1<0或SKIPIF1<0 C．SKIPIF1<0或SKIPIF1<0 D．SKIPIF1<0或SKIPIF1<0二、多選題9．若SKIPIF1<0與SKIPIF1<0為兩條不重合的直線，它們的傾斜角分別是SKIPIF1<0，斜率分別為SKIPIF1<0，則下列命題正確的是(

)A．若斜率SKIPIF1<0，則SKIPIF1<0 B．若SKIPIF1<0，則SKIPIF1<0C．若傾斜角SKIPIF1<0，則SKIPIF1<0 D．若SKIPIF1<0，則SKIPIF1<010．若直線SKIPIF1<0與SKIPIF1<0軸交于點(diǎn)SKIPIF1<0，其傾斜角為SKIPIF1<0，直線SKIPIF1<0繞點(diǎn)SKIPIF1<0順時(shí)針旋轉(zhuǎn)45°后得直線SKIPIF1<0，則直線SKIPIF1<0的傾斜角可能為(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<011．已知點(diǎn)SKIPIF1<0,若過(guò)點(diǎn)SKIPIF1<0的直線SKIPIF1<0與線段SKIPIF1<0相交，則直線SKIPIF1<0的傾斜角可以是(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<012．已知點(diǎn)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0．若SKIPIF1<0為直角三角形，則可能有()A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0三、填空題13．直線SKIPIF1<0的傾斜角是_____