關(guān)于原點(diǎn)對(duì)稱的點(diǎn)的坐標(biāo)課件

關(guān)于原點(diǎn)對(duì)稱的點(diǎn)的坐標(biāo)課件原點(diǎn)對(duì)稱點(diǎn)的定義原點(diǎn)對(duì)稱點(diǎn)的坐標(biāo)規(guī)律原點(diǎn)對(duì)稱點(diǎn)的幾何意義原點(diǎn)對(duì)稱點(diǎn)與函數(shù)圖像的關(guān)系原點(diǎn)對(duì)稱點(diǎn)的計(jì)算方法contents目錄01原點(diǎn)對(duì)稱點(diǎn)的定義定義：如果點(diǎn)$P(x,y)$關(guān)于原點(diǎn)對(duì)稱，則其對(duì)稱點(diǎn)$P'$的坐標(biāo)為$(-x,-y)$。定義原點(diǎn)對(duì)稱的點(diǎn)與原點(diǎn)的距離相等，即$|OP|=|OP'|$。性質(zhì)1原點(diǎn)對(duì)稱的點(diǎn)與原點(diǎn)連線的中點(diǎn)是原點(diǎn)，即線段$OP$與線段$OP'$的中點(diǎn)是原點(diǎn)。性質(zhì)2性質(zhì)點(diǎn)$P(2,3)$關(guān)于原點(diǎn)的對(duì)稱點(diǎn)$P'$的坐標(biāo)為$(-2,-3)$。點(diǎn)$Q(-4,-1)$關(guān)于原點(diǎn)的對(duì)稱點(diǎn)$Q'$的坐標(biāo)為$(4,1)$。舉例舉例2舉例102原點(diǎn)對(duì)稱點(diǎn)的坐標(biāo)規(guī)律規(guī)律一如果點(diǎn)$P(x,y)$關(guān)于原點(diǎn)對(duì)稱，則其對(duì)稱點(diǎn)$P^{prime}(-x,-y)$。規(guī)律二如果點(diǎn)$P(x,y)$關(guān)于原點(diǎn)對(duì)稱，則其對(duì)稱點(diǎn)$P^{prime}(-x,-y)$與原點(diǎn)的距離相等，即$OP=OP^{prime}$。規(guī)律證明一設(shè)點(diǎn)$P(x,y)$關(guān)于原點(diǎn)對(duì)稱，則其對(duì)稱點(diǎn)$P^{prime}(-x,-y)$。根據(jù)兩點(diǎn)之間的距離公式，有$OP=sqrt{x^2+y^2}$，$OP^{prime}=sqrt{(-x)^2+(-y)^2}=sqrt{x^2+y^2}$。因此，$OP=OP^{prime}$。證明二設(shè)點(diǎn)$P(x,y)$關(guān)于原點(diǎn)對(duì)稱，則其對(duì)稱點(diǎn)$P^{prime}(-x,-y)$。根據(jù)向量加法，有$overrightarrow{OP}+overrightarrow{OP^{prime}}=mathbf{0}$，即$(x,y)+(-x,-y)=(0,0)$。因此，點(diǎn)$P^{prime}(-x,-y)$與原點(diǎn)的距離相等。證明在平面幾何中，可以利用原點(diǎn)對(duì)稱的坐標(biāo)規(guī)律來判斷兩點(diǎn)是否關(guān)于原點(diǎn)對(duì)稱。應(yīng)用一在解析幾何中，可以利用原點(diǎn)對(duì)稱的坐標(biāo)規(guī)律來研究曲線和點(diǎn)的對(duì)稱性。應(yīng)用二應(yīng)用03原點(diǎn)對(duì)稱點(diǎn)的幾何意義0102幾何意義例如，點(diǎn)A(x,y)關(guān)于原點(diǎn)的對(duì)稱點(diǎn)B的坐標(biāo)為(-x,-y)。原點(diǎn)對(duì)稱的點(diǎn)在平面直角坐標(biāo)系中表示為兩點(diǎn)，其橫縱坐標(biāo)互為相反數(shù)。圖形表示在坐標(biāo)系中，原點(diǎn)對(duì)稱的點(diǎn)可以用線段連接原點(diǎn)和該點(diǎn)來表示。例如，在平面直角坐標(biāo)系中，點(diǎn)A(2,3)關(guān)于原點(diǎn)的對(duì)稱點(diǎn)B的圖形表示為從原點(diǎn)O(0,0)到點(diǎn)B(-2,-3)的線段。在解析幾何中，原點(diǎn)對(duì)稱的點(diǎn)是重要的概念之一，常用于研究平面圖形的對(duì)稱性質(zhì)。在物理學(xué)、工程學(xué)等領(lǐng)域中，原點(diǎn)對(duì)稱的概念也具有廣泛的應(yīng)用，例如在電路分析、力學(xué)系統(tǒng)等中常常會(huì)涉及到原點(diǎn)對(duì)稱的點(diǎn)的概念。實(shí)際應(yīng)用04原點(diǎn)對(duì)稱點(diǎn)與函數(shù)圖像的關(guān)系如果一個(gè)函數(shù)圖像上的任意一點(diǎn)P(x,y)關(guān)于原點(diǎn)對(duì)稱的點(diǎn)也在該函數(shù)圖像上，則該函數(shù)圖像關(guān)于原點(diǎn)對(duì)稱。函數(shù)圖像關(guān)于原點(diǎn)對(duì)稱如果函數(shù)圖像關(guān)于原點(diǎn)對(duì)稱，則該函數(shù)為奇函數(shù)，滿足f(-x)=-f(x)。函數(shù)圖像關(guān)于原點(diǎn)對(duì)稱的性質(zhì)函數(shù)圖像的對(duì)稱性原點(diǎn)對(duì)稱點(diǎn)在函數(shù)圖像上的表現(xiàn)在函數(shù)圖像上，任意一點(diǎn)P(x,y)關(guān)于原點(diǎn)的對(duì)稱點(diǎn)為(-x,-y)。要點(diǎn)一要點(diǎn)二原點(diǎn)對(duì)稱點(diǎn)與函數(shù)值的關(guān)系如果P(x,y)在函數(shù)圖像上，則(-x,-y)也在該函數(shù)圖像上，且f(x)=f(-x)，f(y)=f(-y)。原點(diǎn)對(duì)稱點(diǎn)與函數(shù)圖像的關(guān)系舉例說明舉例1正弦函數(shù)y=sin(x)的圖像關(guān)于原點(diǎn)對(duì)稱，因?yàn)閷?duì)于任意x，有sin(-x)=-sin(x)。舉例2余弦函數(shù)y=cos(x)的圖像也關(guān)于原點(diǎn)對(duì)稱，因?yàn)閷?duì)于任意x，有cos(-x)=cos(x)。05原點(diǎn)對(duì)稱點(diǎn)的計(jì)算方法設(shè)原點(diǎn)為$(x_0,y_0)$，則關(guān)于原點(diǎn)對(duì)稱的點(diǎn)坐標(biāo)為$(-x_0,-y_0)$。若原點(diǎn)為$(0,0)$，則任意一點(diǎn)$(x,y)$關(guān)于原點(diǎn)的對(duì)稱點(diǎn)坐標(biāo)為$(-x,-y)$。計(jì)算方法1.確定原點(diǎn)的坐標(biāo)。2.根據(jù)原點(diǎn)坐標(biāo)，計(jì)算出對(duì)稱點(diǎn)的橫縱坐標(biāo)。3.得出對(duì)稱點(diǎn)的坐標(biāo)。計(jì)算