第15章軸對(duì)稱(chēng)圖形與等腰三角形15.1軸對(duì)稱(chēng)圖形第2課時(shí)平面直角坐標(biāo)系中的軸對(duì)稱(chēng)知識(shí)點(diǎn)6平面直角坐標(biāo)系中的軸對(duì)稱(chēng)基礎(chǔ)過(guò)關(guān)全練1.(一題多變)(2023安徽淮南謝家集期中)在平面直角坐標(biāo)系

xOy中,點(diǎn)M(-3,2)關(guān)于x軸對(duì)稱(chēng)的點(diǎn)的坐標(biāo)是

()A.(-3,2)

B.(3,2)C.(3,-2)

D.(-3,-2)D解析點(diǎn)M(-3,2)關(guān)于x軸對(duì)稱(chēng)的點(diǎn)的坐標(biāo)為(-3,-2).[變式1]若點(diǎn)P關(guān)于x軸對(duì)稱(chēng)的點(diǎn)M的坐標(biāo)為(4,-5),則點(diǎn)P關(guān)于y

軸對(duì)稱(chēng)的點(diǎn)N的坐標(biāo)為

()A.(-4,5)

B.(4,5)

C.(-4,-5)

D.(-5,4)A解析因?yàn)辄c(diǎn)P關(guān)于x軸對(duì)稱(chēng)的點(diǎn)M的坐標(biāo)為(4,-5),所以點(diǎn)P

的坐標(biāo)是(4,5),所以點(diǎn)P關(guān)于y軸對(duì)稱(chēng)的點(diǎn)N的坐標(biāo)為(-4,5).[變式2]已知點(diǎn)A(a+1,2)關(guān)于y軸的對(duì)稱(chēng)點(diǎn)為B(3,b-1),則a+b的

值為

.-1解析∵點(diǎn)A(a+1,2)關(guān)于y軸的對(duì)稱(chēng)點(diǎn)為B(3,b-1),∴a+1=-3,b-1=2,解得a=-4,b=3,∴a+b=-4+3=-1.2.(教材變式·P126T6)(數(shù)形結(jié)合思想)如圖,在平面直角坐標(biāo)

系中,△ABC的頂點(diǎn)坐標(biāo)分別為A(-1,4),B(-3,1),C(-2,1),網(wǎng)格中

每個(gè)小正方形的邊長(zhǎng)都是1個(gè)單位長(zhǎng)度.(1)畫(huà)出△ABC關(guān)于x軸對(duì)稱(chēng)的△A'B'C',其中點(diǎn)A的對(duì)應(yīng)點(diǎn)是

點(diǎn)A',點(diǎn)B的對(duì)應(yīng)點(diǎn)是點(diǎn)B',點(diǎn)C的對(duì)應(yīng)點(diǎn)是點(diǎn)C';(2)在(1)的條件下,請(qǐng)寫(xiě)出點(diǎn)A',B',C'的坐標(biāo).解析(1)如圖,△A'B'C'即為所求.

(2)由圖可得,A'(-1,-4),B'(-3,-1),C'(-2,-1).能力提升全練3.(2024安徽蕪湖繁昌月考,6,★★☆)若點(diǎn)A(2,5)與點(diǎn)B關(guān)于x

軸對(duì)稱(chēng),點(diǎn)B向右移動(dòng)2個(gè)單位后與點(diǎn)C關(guān)于y軸對(duì)稱(chēng),則點(diǎn)B、

點(diǎn)C的坐標(biāo)分別為

()A.(2,-5),(-2,-5)B.(-2,-5),(-2,5)C.(-2,-5),(2,-5)D.(2,-5),(-4,-5)D解析關(guān)于x軸對(duì)稱(chēng)的兩個(gè)點(diǎn)橫坐標(biāo)相等,縱坐標(biāo)互為相反

數(shù),因?yàn)锳(2,5),所以點(diǎn)B的坐標(biāo)為(2,-5).點(diǎn)B向右平移2個(gè)單位

后的坐標(biāo)為(4,-5).關(guān)于y軸對(duì)稱(chēng)的兩個(gè)點(diǎn),橫坐標(biāo)互為相反數(shù),

縱坐標(biāo)相等,所以點(diǎn)C的坐標(biāo)為(-4,-5).4.(2024安徽六安舒城六校聯(lián)考,5,★★☆)紀(jì)運(yùn)同學(xué)誤將點(diǎn)A

的橫、縱坐標(biāo)次序顛倒,寫(xiě)成A(3,5).舒林同學(xué)誤將點(diǎn)B的坐

標(biāo)寫(xiě)點(diǎn)B成關(guān)于y軸對(duì)稱(chēng)的點(diǎn)的坐標(biāo),即B(-5,-3),則下列說(shuō)法

正確的是

()A.A、B兩點(diǎn)關(guān)于x軸對(duì)稱(chēng)B.A、B兩點(diǎn)關(guān)于y軸對(duì)稱(chēng)C.直線AB與x軸平行D.點(diǎn)A和點(diǎn)B重合A解析因?yàn)辄c(diǎn)A的橫、縱坐標(biāo)次序顛倒,所以點(diǎn)A的正確坐標(biāo)

為(5,3),點(diǎn)B的坐標(biāo)誤寫(xiě)成點(diǎn)B關(guān)于y軸對(duì)稱(chēng)的點(diǎn)的坐標(biāo),所以

點(diǎn)B的正確坐標(biāo)為(5,-3).所以A、B兩點(diǎn)關(guān)于x軸對(duì)稱(chēng).5.(2023安徽合肥廬江期中,2,★★☆)若點(diǎn)A(a-1,2021)與點(diǎn)B

(2022,b-1)關(guān)于y軸對(duì)稱(chēng),則(a+b)2023=

()A.1

B.-1

C.-2021

D.2022A解析∵點(diǎn)A(a-1,2021)與點(diǎn)B(2022,b-1)關(guān)于y軸對(duì)稱(chēng),∴a-1

=-2022,b-1=2021,∴a=1-2022=-2021,b=1+2021=2022,∴(a

+b)2023=12023=1.6.(2024安徽黃山徽州月考,12,★★☆)在平面直角坐標(biāo)系中,

已知點(diǎn)A關(guān)于x軸對(duì)稱(chēng)的點(diǎn)B的坐標(biāo)是(3-5m,2-m),點(diǎn)B是第二

象限的整點(diǎn)(橫、縱坐標(biāo)都是整數(shù)的點(diǎn),稱(chēng)之為整點(diǎn)),則點(diǎn)A

關(guān)于y軸對(duì)稱(chēng)的點(diǎn)C的坐標(biāo)是

.(2,-1)解析∵點(diǎn)B在第二象限,∴

∴

<m<2.∵3-5m,2-m都是整數(shù),∴m=1,∴3-5m=-2,2-m=1,∴點(diǎn)B的坐標(biāo)為(-2,1),∵

點(diǎn)A與點(diǎn)B關(guān)于x軸對(duì)稱(chēng),∴點(diǎn)A的坐標(biāo)為(-2,-1),∴點(diǎn)A關(guān)于y軸

對(duì)稱(chēng)的點(diǎn)C的坐標(biāo)為(2,-1).7.(2020四川達(dá)州中考,12,★★☆)如圖,點(diǎn)P(-2,1)與點(diǎn)Q(a,b)

關(guān)于直線l:y=-1對(duì)稱(chēng),則a+b=

.-5解析∵點(diǎn)P到直線l的距離為1-(-1)=2,點(diǎn)P(-2,1)與點(diǎn)Q(a,b)

關(guān)于直線l:y=-1對(duì)稱(chēng),∴點(diǎn)Q到直線l的距離也為2,∴點(diǎn)Q到x軸

的距離為1+2=3,∴a=-2,b=-3,∴a+b=-2-3=-5.8.(新考向·尺規(guī)作圖)(2024安徽六安金寨期末,15,★★☆)如

圖,△ABC的三個(gè)頂點(diǎn)的坐標(biāo)分別為A(1,2),B(3,1),C(2,4).(1)在平面直角坐標(biāo)系中作出△ABC關(guān)于y軸對(duì)稱(chēng)的△A'B'C',

并寫(xiě)出點(diǎn)A的對(duì)應(yīng)點(diǎn)A'的坐標(biāo);(2)在(1)的條件下,求△CBB'的面積.解析(1)如圖,△A'B'C'即為所求.

根據(jù)圖可知,A'(-1,2).(2)S△CBB'=

×6×3=9.9.(2024遼寧鞍山期末改編,13,★★☆)(1)如圖,點(diǎn)A的坐標(biāo)為(-1,2),點(diǎn)B的坐標(biāo)為(-3,-2),在y軸上存在一點(diǎn)P,滿足AP+BP的值

最小,求點(diǎn)P的坐標(biāo);(2)已知點(diǎn)A(0,1),B(2,1),點(diǎn)P為x軸上一點(diǎn),當(dāng)PA+PB最小時(shí),求

點(diǎn)P的坐標(biāo).解析(1)如圖1,作點(diǎn)B關(guān)于y軸的對(duì)稱(chēng)點(diǎn)B',連接AB',交y軸于

點(diǎn)P,則此時(shí)AP+PB的值最小,為AB'的長(zhǎng).∵點(diǎn)B的坐標(biāo)為(-3,-

2),∴B'(3,-2).設(shè)直線AB'的解析式為y=kx+b(k≠0),將A(-1,2),B'

(3,-2)代入,得

∴

∴y=-x+1.當(dāng)x=0時(shí),y=1,∴P(0,1).

(2)如圖2,作點(diǎn)A關(guān)于x軸的對(duì)稱(chēng)點(diǎn)A',連接A'B交x軸于點(diǎn)P,此

時(shí)PA+PB的值最小,過(guò)B作BH⊥x軸于點(diǎn)H.∵點(diǎn)A(0,1),B(2,1),∴A'(0,-1),H(2,0),∴OA'=BH=1,OH=2.在△

A'OP與△BHP中,

∴△A'OP≌△BHP(AAS),∴OP=PH.∵OH=2,∴OP=PH=

OH=1,∴點(diǎn)P的坐標(biāo)是(1,0).素養(yǎng)探究全練10.(幾何直觀)由邊長(zhǎng)為1的若干個(gè)小正方形拼成的網(wǎng)格圖如

圖所示,△ABC的頂點(diǎn)A,B,C均在小正方形的頂點(diǎn)上.(1)在圖中建立恰當(dāng)?shù)钠矫嬷苯亲鴺?biāo)系,取小正方形的邊長(zhǎng)為

一個(gè)單位長(zhǎng)度,使點(diǎn)A的坐標(biāo)為(2,1),并寫(xiě)出B,C兩點(diǎn)的坐標(biāo);(2)在(1)中建立的平面直角坐標(biāo)系中畫(huà)出△ABC關(guān)于x軸對(duì)

稱(chēng)的△A'B'C';(3)求△ABC的面積.解析(1)建立的平面直角坐標(biāo)系如圖.

B(5,-1),C(3,-2).(2)如圖所示,△A'B'C'即為所求.(3)S△ABC=3×3-

×1×3-

×1×2-

×2×3=

.11.(推理能力)已知長(zhǎng)方形紙片ABCD中,點(diǎn)E,F,G分別在邊

AB,DA,BC上,將三角形AEF沿EF翻折,點(diǎn)A落在點(diǎn)A'處,將三

角形EBG沿EG翻折,點(diǎn)B落在點(diǎn)B'處.(1)如圖①,點(diǎn)E,A',B'共線時(shí),求∠GEF的度數(shù);(2)如圖②③,點(diǎn)E,A',B'不共線時(shí),設(shè)∠GEF=α,∠A'EB'=β,請(qǐng)分

別寫(xiě)出圖②③中α,β滿足的數(shù)量關(guān)系,并說(shuō)明理由.圖①圖②圖③解析(1)由翻折得∠AEF=∠A'EF,∠BEG=∠B'EG,∴2∠B'

EF+2∠B'EG=180°,∴∠GEF=∠B'EF+∠B'EG=90°.(2)題圖②:2α-β=180°.理由:由翻折得∠AEF=∠A'EF,∠BEG=∠B'EG.∵∠GEF=α,

∠A'EB'=β,2∠A'EF+2∠B'EG+∠A'EB'=180°,∴2(