能力提升精練課件：3 軸對(duì)稱(chēng)與坐標(biāo)變化

第三章位置與坐標(biāo)3軸對(duì)稱(chēng)與坐標(biāo)變化能力提升精練3軸對(duì)稱(chēng)與坐標(biāo)變化1.將平面直角坐標(biāo)系中某個(gè)圖形上的各個(gè)點(diǎn)的橫坐標(biāo)都乘以-1,縱坐標(biāo)不變,所得圖形與原圖形的位置關(guān)系是(

)

A.關(guān)于x軸對(duì)稱(chēng) B.關(guān)于y軸對(duì)稱(chēng)C.關(guān)于原點(diǎn)對(duì)稱(chēng) D.重合【解析】

因?yàn)楦鱾€(gè)點(diǎn)的縱坐標(biāo)不變,橫坐標(biāo)都乘以-1,即縱坐標(biāo)相同,橫坐標(biāo)互為相反數(shù),所以所得圖形與原圖形關(guān)于y軸對(duì)稱(chēng).故選B.2.如圖,在3×3的正方形網(wǎng)格中有四個(gè)格點(diǎn)A,B,C,D,以其中一點(diǎn)為坐標(biāo)原點(diǎn),以互相垂直的網(wǎng)格線(xiàn)所在直線(xiàn)為坐標(biāo)軸,建立平面直角坐標(biāo)系,使其余三個(gè)點(diǎn)中存在兩個(gè)點(diǎn)關(guān)于一條坐標(biāo)軸對(duì)稱(chēng),則坐標(biāo)原點(diǎn)應(yīng)選(

)A.A點(diǎn)

B.B點(diǎn)

C.C點(diǎn)

D.D點(diǎn)【解析】當(dāng)以點(diǎn)B為坐標(biāo)原點(diǎn),以互相垂直的網(wǎng)格線(xiàn)所在直線(xiàn)為坐標(biāo)軸,建立平面直角坐標(biāo)系時(shí),點(diǎn)A,C關(guān)于一條坐標(biāo)軸對(duì)稱(chēng).故選B.3.[2021湖北武漢第六初級(jí)中學(xué)期中]如圖,在平面直角坐標(biāo)系中,對(duì)△ABC進(jìn)行循環(huán)往復(fù)的軸對(duì)稱(chēng)變換,若原來(lái)點(diǎn)A的坐標(biāo)是(1,2),則經(jīng)過(guò)第2021次變換后點(diǎn)A的對(duì)應(yīng)點(diǎn)的坐標(biāo)為(

)A.(1,-2) B.(-1,-2)C.(-1,2) D.(1,2)【解析】

點(diǎn)A第1次關(guān)于y軸對(duì)稱(chēng)的對(duì)應(yīng)點(diǎn)的坐標(biāo)為(-1,2),第2次關(guān)于x軸對(duì)稱(chēng)的對(duì)應(yīng)點(diǎn)的坐標(biāo)為(-1,-2),第3次關(guān)于y軸對(duì)稱(chēng)的對(duì)應(yīng)點(diǎn)的坐標(biāo)為(1,-2),第4次關(guān)于x軸對(duì)稱(chēng)的對(duì)應(yīng)點(diǎn)的坐標(biāo)為(1,2)······所以每4次軸對(duì)稱(chēng)變換為一個(gè)循環(huán).因?yàn)?021÷4=505······1,所以經(jīng)過(guò)第2021次變換后點(diǎn)A的對(duì)應(yīng)點(diǎn)的坐標(biāo)為(-1,2).故選C.4.若點(diǎn)P(a,b)關(guān)于y軸的對(duì)稱(chēng)點(diǎn)是P1,而點(diǎn)P1關(guān)于x軸的對(duì)稱(chēng)點(diǎn)是P2,已知點(diǎn)P2的坐標(biāo)為(-3,4),則a=

,b=

.

【解析】因?yàn)辄c(diǎn)P1和P2(-3,4)關(guān)于x軸對(duì)稱(chēng),所以點(diǎn)P1的坐標(biāo)為(-3,-4),又因?yàn)辄c(diǎn)P(a,b)和點(diǎn)P1(-3,-4)關(guān)于y軸對(duì)稱(chēng),所以點(diǎn)P的坐標(biāo)為(3,-4),所以a=3,b=-4.

平面直角坐標(biāo)系中對(duì)稱(chēng)點(diǎn)的坐標(biāo)特征(1)關(guān)于x軸對(duì)稱(chēng):橫坐標(biāo)相同,縱坐標(biāo)互為相反數(shù),即點(diǎn)P(a,b)關(guān)于x軸對(duì)稱(chēng)的點(diǎn)是(a,-b).(2)關(guān)于y軸對(duì)稱(chēng):橫坐標(biāo)互為相反數(shù),縱坐標(biāo)相同,即點(diǎn)P(a,b)關(guān)于y軸對(duì)稱(chēng)的點(diǎn)是(-a,b).(3)關(guān)于坐標(biāo)原點(diǎn)對(duì)稱(chēng):橫、縱坐標(biāo)都互為相反數(shù),即點(diǎn)P(a,b)關(guān)于坐標(biāo)原點(diǎn)對(duì)稱(chēng)的點(diǎn)是(-a,-b).注意:平面直角坐標(biāo)系中,任何一個(gè)點(diǎn)都存在關(guān)于x軸、y軸、原點(diǎn)對(duì)稱(chēng)的點(diǎn).對(duì)稱(chēng)點(diǎn)的坐標(biāo)變化也可以這樣理解:①關(guān)于x軸對(duì)稱(chēng)的點(diǎn),橫坐標(biāo)乘1,縱坐標(biāo)乘-1;②關(guān)于y軸對(duì)稱(chēng)的點(diǎn),縱坐標(biāo)乘1,橫坐標(biāo)乘-1;③關(guān)于原點(diǎn)對(duì)稱(chēng)的點(diǎn),橫坐標(biāo)、縱坐標(biāo)都乘-1.歸納總結(jié)5.如圖,正五邊形ABCDE放入某平面直角坐標(biāo)系后,若頂點(diǎn)A,B,C,D的坐標(biāo)分別是(0,a),(-3,2),(b,m),(c,m),則點(diǎn)E的坐標(biāo)是

.

【解析】因?yàn)辄c(diǎn)A的坐標(biāo)為(0,a),所以點(diǎn)A在該平面直角坐標(biāo)系的y軸上,因?yàn)辄c(diǎn)C,D的坐標(biāo)分別為(b,m),(c,m),所以CD∥x軸,因?yàn)檎暹呅蜛BCDE是軸對(duì)稱(chēng)圖形,所以該平面直角坐標(biāo)系經(jīng)過(guò)點(diǎn)A的y軸是正五邊形ABCDE的一條對(duì)稱(chēng)軸,所以點(diǎn)B,E關(guān)于y軸對(duì)稱(chēng),因?yàn)辄c(diǎn)B的坐標(biāo)為(-3,2),所以點(diǎn)E的坐標(biāo)為(3,2).6.如圖,已知等邊三角形ABC的邊長(zhǎng)為3,寫(xiě)出點(diǎn)A,B,C的坐標(biāo),并畫(huà)出△ABC關(guān)于y軸對(duì)稱(chēng)的圖形△A1B1C1.

7.[2021河南鄭州桐柏一中期中]如圖,在平面直角坐標(biāo)系中,△ABC的三個(gè)頂點(diǎn)的坐標(biāo)分別是A(1,3),B(4,1),C(-2,-2).(1)在圖中作出△ABC關(guān)于y軸對(duì)稱(chēng)的圖形△A1B1C1,并寫(xiě)出A1,B1,C1三點(diǎn)的坐標(biāo);(2)在x軸上有一點(diǎn)P,畫(huà)圖找出使得PA+PB取得最小值時(shí)點(diǎn)P的位置.(不寫(xiě)作法,保留作圖痕跡)【解析】

(1)如圖,△A1B1C1即所求.點(diǎn)A1的坐標(biāo)為(-1,3),點(diǎn)B1的坐標(biāo)為(-4,1),點(diǎn)C1的坐標(biāo)為(2,-2).(2)如圖,作點(diǎn)B關(guān)于x軸的對(duì)稱(chēng)點(diǎn)B',連接AB'交x軸于點(diǎn)P,點(diǎn)P即所求.8.[2020湖北恩施州期末]如圖,在平面直角坐標(biāo)系中,直線(xiàn)l過(guò)點(diǎn)M(3,0),且平行于y軸.(1)如果△ABC三個(gè)頂點(diǎn)的坐標(biāo)分別是A(-2,0),B(-1,0),C(-1,2),△ABC關(guān)于y軸的對(duì)稱(chēng)圖形是△A1B1C1,△A1B1C1關(guān)于直線(xiàn)l的對(duì)稱(chēng)圖形是△A2B2