一个有关长方形的结论的妙用湖北省襄阳市襄州区黄集镇初级中学 张昌林#TRS_AUTOADD_1314171186668 {MARGIN-TOP: 0px; MARGIN-BOTTOM: 0px}#TRS_AUTOADD_1314171186668 P {MARGIN-TOP: 0px; MARGIN-BOTTOM: 0px}#TRS_AUTOADD_1314171186668 TD {MARGIN-TOP: 0px; MARGIN-BOTTOM: 0px}#TRS_AUTOADD_1314171186668 DIV {MARGIN-TOP: 0px; MARGIN-BOTTOM: 0px}#TRS_AUTOADD_1314171186668 LI {MARGIN-TOP: 0px; MARGIN-BOTTOM: 0px}/**---JSON--{"":{"margin-top":"0","margin-bottom":"0"},"p":{"margin-top":"0","margin-bottom":"0"},"td":{"margin-top":"0","margin-bottom":"0"},"div":{"margin-top":"0","margin-bottom":"0"},"li":{"margin-top":"0","margin-bottom":"0"}}--**/DIV.MyFav_1314170907054 P.MsoNormal{TEXT-JUSTIFY: inter-ideograph; FONT-SIZE: 10.5pt; MARGIN: 0cm 0cm 0pt; FONT-FAMILY: "Times New Roman"; TEXT-ALIGN: justify}DIV.MyFav_1314170907054 LI.MsoNormal{TEXT-JUSTIFY: inter-ideograph; FONT-SIZE: 10.5pt; MARGIN: 0cm 0cm 0pt; FONT-FAMILY: "Times New Roman"; TEXT-ALIGN: justify}DIV.MyFav_1314170907054 DIV.MsoNormal{TEXT-JUSTIFY: inter-ideograph; FONT-SIZE: 10.5pt; MARGIN: 0cm 0cm 0pt; FONT-FAMILY: "Times New Roman"; TEXT-ALIGN: justify}DIV.MyFav_1314170907054 P.MsoHeader{BORDER-RIGHT: medium none; PADDING-RIGHT: 0cm; BORDER-TOP: medium none; PADDING-LEFT: 0cm; FONT-SIZE: 9pt; PADDING-BOTTOM: 0cm; MARGIN: 0cm 0cm 0pt; BORDER-LEFT: medium none; LAYOUT-GRID-MODE: char; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; FONT-FAMILY: "Times New Roman"; TEXT-ALIGN: center}DIV.MyFav_1314170907054 LI.MsoHeader{BORDER-RIGHT: medium none; PADDING-RIGHT: 0cm; BORDER-TOP: medium none; PADDING-LEFT: 0cm; FONT-SIZE: 9pt; PADDING-BOTTOM: 0cm; MARGIN: 0cm 0cm 0pt; BORDER-LEFT: medium none; LAYOUT-GRID-MODE: char; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; FONT-FAMILY: "Times New Roman"; TEXT-ALIGN: center}DIV.MyFav_1314170907054 DIV.MsoHeader{BORDER-RIGHT: medium none; PADDING-RIGHT: 0cm; BORDER-TOP: medium none; PADDING-LEFT: 0cm; FONT-SIZE: 9pt; PADDING-BOTTOM: 0cm; MARGIN: 0cm 0cm 0pt; BORDER-LEFT: medium none; LAYOUT-GRID-MODE: char; PADDING-TOP: 0cm; BORDER-BOTTOM: medium none; FONT-FAMILY: "Times New Roman"; TEXT-ALIGN: center}DIV.MyFav_1314170907054 P.MsoFooter{FONT-SIZE: 9pt; MARGIN: 0cm 0cm 0pt; LAYOUT-GRID-MODE: char; FONT-FAMILY: "Times New Roman"}DIV.MyFav_1314170907054 LI.MsoFooter{FONT-SIZE: 9pt; MARGIN: 0cm 0cm 0pt; LAYOUT-GRID-MODE: char; FONT-FAMILY: "Times New Roman"}DIV.MyFav_1314170907054 DIV.MsoFooter{FONT-SIZE: 9pt; MARGIN: 0cm 0cm 0pt; LAYOUT-GRID-MODE: char; FONT-FAMILY: "Times New Roman"}DIV.MyFav_1314170907054 A:link{COLOR: #2200cc; FONT-FAMILY: Arial; TEXT-DECORATION: underline}DIV.MyFav_1314170907054 SPAN.MsoHyperlink{COLOR: #2200cc; FONT-FAMILY: Arial; TEXT-DECORATION: underline}DIV.MyFav_1314170907054 A:visited{COLOR: purple; TEXT-DECORATION: underline}DIV.MyFav_1314170907054 SPAN.MsoHyperlinkFollowed{COLOR: purple; TEXT-DECORATION: underline}DIV.MyFav_1314170907054 DIV.Section1{page: Section1}一类有关反比例函数的题目,要用到一个有关长方形的结论来解显得极其容易,若对这个结论没掌握好要解这类题目是不容易的,下面我们来一起学习一下.

结论1:如图1,长方形ABCD的对角线把长方形分成面积相等的两部分.
利用三角形全等容易证明.
结论2: :如图2,AC是长方形ABCD的对角线,点E是对角线AC上一动点,过点E分别做AB、AD的平行线段IF、HG,点I、F分别在AD、BC上,点H、G分别在AB、DC上.则图中阴影部分的面积相等即.
证明:如图2,在长方形ABCD中,由结论1知①.
同理在长方形AHEI中,由结论1知②.
同理在长方形EFGC中,由结论1知③.
①-②-③得:.
例1(2011年浙江省杭州市城南初级中学中考数学模拟试题)如图3,矩形ABCD的对角线BD经过坐标原点,矩形的边分别平行于坐标轴,点C在反比例函数的图象上.若点A的坐标为(-2,-2),则k的值为( )
图3
A.-2 B.2 C.3 D.4
点评 由结论2,易知k=4,答案:D
例2(2011甘肃兰州,15,4分)如图4,矩形ABCD的对角线BD经过坐标原点,矩形的边分别平行于坐标轴,点C在反比例函数的图象上.若点A的坐标为(-2,-2),则k的值为( )
图4
A.1 B.-3 C.4 D.1或-3
点评 由结论2,易知,解得:k=1或-3 答案:D
以上两例说明了这个结论在解题中的重要作用,若平时对这个结论不熟悉要解出这两道例题显然是极其不容易的.
2011-08-24 人教网
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