直線方程中的對(duì)稱(chēng)問(wèn)題課件

對(duì)稱(chēng)問(wèn)題

對(duì)稱(chēng)問(wèn)題對(duì)稱(chēng)問(wèn)題中心對(duì)稱(chēng)問(wèn)題點(diǎn)關(guān)于于點(diǎn)的對(duì)稱(chēng)線關(guān)于點(diǎn)的對(duì)稱(chēng)軸對(duì)稱(chēng)問(wèn)題點(diǎn)關(guān)于線的對(duì)稱(chēng)線關(guān)于線的對(duì)稱(chēng)對(duì)稱(chēng)問(wèn)題中心對(duì)稱(chēng)問(wèn)題點(diǎn)關(guān)于于點(diǎn)的對(duì)稱(chēng)線關(guān)于點(diǎn)的對(duì)稱(chēng)軸對(duì)稱(chēng)問(wèn)題1.點(diǎn)關(guān)于點(diǎn)對(duì)稱(chēng)①點(diǎn)關(guān)于點(diǎn)對(duì)稱(chēng)：說(shuō)明兩點(diǎn)P和Q關(guān)于點(diǎn)M對(duì)稱(chēng)的幾何特征PMQ點(diǎn)M是線段PQ的中點(diǎn)1.點(diǎn)關(guān)于點(diǎn)對(duì)稱(chēng)①點(diǎn)關(guān)于點(diǎn)對(duì)稱(chēng)：說(shuō)明兩點(diǎn)P和Q關(guān)于點(diǎn)M對(duì)稱(chēng)點(diǎn)關(guān)于點(diǎn)的對(duì)稱(chēng)注：解題要點(diǎn)：中點(diǎn)公式的運(yùn)用點(diǎn)關(guān)于點(diǎn)的對(duì)稱(chēng)注：解題要點(diǎn)：中點(diǎn)公式的運(yùn)用例1.已知點(diǎn)A(5,8)，B(-4，1)，試求A點(diǎn)關(guān)于B點(diǎn)的對(duì)稱(chēng)點(diǎn)C的坐標(biāo)。知識(shí)運(yùn)用與解題研究一、點(diǎn)關(guān)于點(diǎn)對(duì)稱(chēng)解題要點(diǎn)：中點(diǎn)公式的運(yùn)用ACBxyOC(-13，-6)-4=5+x21=8+y2解:設(shè)C(x,y)則得x=-13y=-6∴···例1.已知點(diǎn)A(5,8)，B(-4，1)，試求A點(diǎn)知特殊的,P(x,y)關(guān)于下列直線的對(duì)稱(chēng)點(diǎn):2、軸對(duì)稱(chēng):①點(diǎn)關(guān)于線對(duì)稱(chēng):關(guān)于X軸的對(duì)稱(chēng)點(diǎn)

___________關(guān)于Y軸的對(duì)稱(chēng)點(diǎn)___________關(guān)于直線y=x的對(duì)稱(chēng)點(diǎn)____________關(guān)于直線y=-x的對(duì)稱(chēng)點(diǎn)_____________(x,-y)(-x,y)(y,x)(-y,-x)特殊的,P(x,y)關(guān)于下列直線的對(duì)稱(chēng)點(diǎn):2、軸對(duì)稱(chēng)3點(diǎn)關(guān)于直線的對(duì)稱(chēng)點(diǎn)PQl垂直中點(diǎn)O說(shuō)明兩點(diǎn)P和Q關(guān)于直線l對(duì)稱(chēng)的幾何特征∟3點(diǎn)關(guān)于直線的對(duì)稱(chēng)點(diǎn)PQl垂直中點(diǎn)O說(shuō)明兩點(diǎn)P和Q關(guān)于直直線l是線段PQ的垂直平分線，即xyOPQl：ax+by+c=0，a≠0,b≠0,點(diǎn)P(x0,y0),如何求點(diǎn)P關(guān)于直線l的對(duì)稱(chēng)點(diǎn)Q的坐標(biāo)？

1.線段PQ的中點(diǎn)在直線l上，2.線段PQ和直線l垂直問(wèn)題研討直線l是線段PQ的垂直平分線，即xyOPQl：ax+by+c練一練、求點(diǎn)A(3,2)關(guān)于直線ｌ:y=2x+1的對(duì)稱(chēng)點(diǎn)B的坐標(biāo).解:設(shè)B點(diǎn)坐標(biāo)為(a,b),因ｌ垂直平分線段AB,則解得a=–1,b=4.ｌA(3,2)Oxy所以所求點(diǎn)B的坐標(biāo)為（－1，4）.B(a,b)練一練、求點(diǎn)A(3,2)關(guān)于直線ｌ:y=2x+1的對(duì)稱(chēng)點(diǎn)B（法一）：∵直線AB⊥l,直線AB過(guò)點(diǎn)（-7，1）∴直線AB的方程為y-1=-（x+7）即x+2y+5=0由解得即AB的中點(diǎn)為（1，-3），又A（-7，1）由中點(diǎn)坐標(biāo)公式得B的坐標(biāo)為（9，-7）.例.

求點(diǎn)A(-7,1)關(guān)于直線l:2x-y-5=0的對(duì)稱(chēng)點(diǎn)B的坐標(biāo).（法一）：∵直線AB⊥l,直線AB過(guò)點(diǎn)（-7，1）∴直線例.求點(diǎn)A(-7,1)關(guān)于直線l:2x-y-5=0的對(duì)稱(chēng)點(diǎn)B的坐標(biāo).（法二）：設(shè)B(m,n）由點(diǎn)關(guān)于直線對(duì)稱(chēng)的定義知:

線段AB⊥l

即;=-1①線段AB被直線l平分,即線段AB的中點(diǎn)在直線l上,故有

2--5=0②聯(lián)立①②解得m=9n=-7∴B(9,-7)例.求點(diǎn)A(-7,1)關(guān)于直線l:2x-y-5=0的對(duì)稱(chēng)點(diǎn)B(法三)設(shè)B(m,n）由點(diǎn)關(guān)于直線對(duì)稱(chēng)的定義知:

線段AB⊥l

即;=-1①由題知：A，B兩點(diǎn)關(guān)于直線l對(duì)稱(chēng)，則A,B兩點(diǎn)到直線l的距離是相等的，則：

聯(lián)立①②解得m=9n=-7∴B(9,-7)例.求點(diǎn)A(-7,1)關(guān)于直線l:2x-y-5=0的對(duì)稱(chēng)點(diǎn)B的坐標(biāo)(法三)設(shè)B(m,n）由點(diǎn)關(guān)于直線對(duì)稱(chēng)的定義知:線段點(diǎn)關(guān)于直線的對(duì)稱(chēng)

點(diǎn)關(guān)于直線的對(duì)稱(chēng)

②線關(guān)于點(diǎn)的對(duì)稱(chēng)問(wèn)題：ABPDCmn說(shuō)明兩直線關(guān)于一點(diǎn)對(duì)稱(chēng)的幾何特征②線關(guān)于點(diǎn)的對(duì)稱(chēng)問(wèn)題：ABPDCmn說(shuō)明兩直線關(guān)于一點(diǎn)對(duì)稱(chēng)

對(duì)稱(chēng)中心到兩平行直線的距離相等，利用平行直線系。

直線m上任取兩點(diǎn)A、B求出n上的對(duì)稱(chēng)點(diǎn)C、D后，由兩點(diǎn)確定一條直線

n上的任意一點(diǎn)的對(duì)稱(chēng)點(diǎn)在m上，利用相關(guān)點(diǎn)法。方法小結(jié)對(duì)稱(chēng)中心到兩平行直線的距離相等，利用平行直線系P(2,–1)xyOy=3x–4

求直線y=3x–4關(guān)于點(diǎn)P(2,–1)的對(duì)稱(chēng)直線方程.（二）直線關(guān)于點(diǎn)的對(duì)稱(chēng)

P(2,–1)xyOy=3x–4求直線y=3x–4關(guān)P(2,–1)xyOy=3x–4求直線y=3x–4關(guān)于點(diǎn)P(2,–1)的對(duì)稱(chēng)直線方程.

P(2,–1)xyOy=3x–4求直線y=3x–4關(guān)于點(diǎn)P(

求直線y=3x–4關(guān)于點(diǎn)P(2,–1)的對(duì)稱(chēng)直線方程.P(2,–1)xyOy=3x–4法三：分析一:將直線的對(duì)稱(chēng)轉(zhuǎn)化為直線上的點(diǎn)的對(duì)稱(chēng).求直線y=3x–4關(guān)于點(diǎn)P(2,–1)的對(duì)稱(chēng)直線方程直線關(guān)于點(diǎn)對(duì)稱(chēng)

法二：利用點(diǎn)到直線的距離l

1//l

2且P到兩直線

等距。主要方法：法一：轉(zhuǎn)化成求點(diǎn)關(guān)于點(diǎn)的對(duì)稱(chēng)直線關(guān)于點(diǎn)對(duì)稱(chēng)

法二：利用點(diǎn)到直線的距離l1//l例.試求直線l1:x-y-2=0關(guān)于直線l2:3x-y+3=0對(duì)稱(chēng)的直線l的方程。解題要點(diǎn)：由線關(guān)于線對(duì)稱(chēng)轉(zhuǎn)化為點(diǎn)關(guān)于點(diǎn)對(duì)稱(chēng)思考：若l1//l2,如何求l1關(guān)于l2的對(duì)稱(chēng)直線方程？C1lC2M(x,y)M＇

(x1,y1)l1l2l1'（二）直線關(guān)于直線的對(duì)稱(chēng)例.試求直線l1:x-y-2=0關(guān)于直線l2:3x-y+練一練：求直線3x-2y+6=0關(guān)于直線x-2y+1=0的對(duì)稱(chēng)的直線方程。XYO