平行四边形面积推导与演示（教材四）
2013-05-06 10:14
本GeoGebra范例用于演示和推导平行四边形的面积公式。通过几何作图命令构建平行四边形及其相关辅助图形，直观展示平行四边形与矩形之间的面积转化关系。适用于数学教学，帮助学生通过动态几何图形深入理解平行四边形面积的计算原理（底乘高），是几何面积教学的重要互动课件。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="4.0" id="c49d4899-2a01-4286-bcc4-4778cd59d829" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1055" height="686" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.24170616113744076" orientation="1" />
</panes>
<views>
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="200" window="0,0,400,400" />
<view id="1" visible="true" inframe="false" stylebar="false" location="1" size="1047" window="0,0,400,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="255" window="0,0,200,400" />
<view id="16" visible="false" inframe="true" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="32" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" visible="false" inframe="true" stylebar="false" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 39 59 || 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 || 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 || 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 , 14 66 68 | 25 52 60 61 || 40 41 42 , 27 28 35 , 6" />
<input show="true" cmd="true" top="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
<graphicsSettings javaLatexFonts="false"/>
<consProtColumns col0="true" col1="true" col2="false" col3="true" col4="false" col5="true" col6="true" col7="false"/>
<consProtocol useColors="true" addIcons="false" showOnlyBreakpoints="false"/>
</gui>
<euclidianView>
<size width="1047" height="523"/>
<coordSystem xZero="484.3999999999998" yZero="222.9" scale="54.99999999999995" yscale="54.99999999999998"/>
<evSettings axes="false" grid="true" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="13" gridType="0"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="10"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<kernel>
<continuous val="false"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0"/>
<coordStyle val="0"/>
<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="Area of parallelograms" author="David Taub" date="">
<worksheetText above="Both parallelograms start the same size and shape. You can drag points B and G to change their shape. Compare what happens to their area as you change the shape? Why is this happening? What is the difference between the two?" below=""/>
<element type="point" label="A">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-21" y="21"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="-1.0" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="A" a1="2"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eigenvectors x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
<matrix A0="1.0" A1="1.0" A2="-3.0" A3="0.0" A4="1.0" A5="-0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="true" label="true"/>
<objColor r="0" g="51" b="204" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-26" y="3"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="-0.9999999999999999" y="2.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="point" label="C">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="8" y="17"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="3.0" y="0.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="A" a1="B"/>
<output a0="a"/>
</command>
<element type="segment" label="a">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-40" y="7"/>
<labelMode val="2"/>
<coords x="-2.0" y="1.1102230246251565E-16" z="-2.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A" a1="C"/>
<output a0="b"/>
</command>
<element type="segment" label="b">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-4" y="1"/>
<labelMode val="2"/>
<coords x="0.0" y="4.0" z="-0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Line">
<input a0="C" a1="a"/>
<output a0="d"/>
</command>
<element type="line" label="d">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.0" y="1.1102230246251565E-16" z="6.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Line">
<input a0="B" a1="b"/>
<output a0="e"/>
</command>
<element type="line" label="e">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="4.0" z="-8.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="d" a1="e"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-5" y="-1"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="24.0" y="16.0" z="8.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="B" a1="D"/>
<output a0="f"/>
</command>
<element type="segment" label="f">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-4" y="-26"/>
<labelMode val="2"/>
<coords x="0.0" y="4.0" z="-8.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="D" a1="C"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="27" y="0"/>
<labelMode val="2"/>
<coords x="2.0" y="0.0" z="-6.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Polygon">
<input a0="B" a1="D" a2="C" a3="A"/>
<output a0="poly1" a1="b_1" a2="d_1" a3="c_1" a4="a_1"/>
</command>
<element type="polygon" label="poly1">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="-8" y="0"/>
<labelMode val="2"/>
</element>
<element type="segment" label="b_1">
<show object="true" label="true"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-6" y="-29"/>
<labelMode val="2"/>
<coords x="0.0" y="4.0" z="-8.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="d_1">
<show object="true" label="true"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="27" y="5"/>
<labelMode val="2"/>
<coords x="2.0" y="0.0" z="-6.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="c_1">
<show object="true" label="true"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-6" y="34"/>
<labelMode val="2"/>
<coords x="0.0" y="-4.0" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="a_1">
<show object="true" label="true"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-37" y="2"/>
<labelMode val="2"/>
<coords x="-2.0" y="1.1102230246251565E-16" z="-2.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="Textpoly1" exp=""Area " + (Name[B]) + (Name[D]) + (Name[C]) + (Name[A]) + " = " + poly1" />
<element type="text" label="Textpoly1">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" size="4" style="1"/>
<absoluteScreenLocation x="472" y="77"/>
</element>
<element type="point" label="E">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-29" y="18"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="-1.0" y="-5.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="point" label="F">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="8" y="15"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="3.0" y="-5.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="line" label="h">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="0.0" y="1.0" z="3.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="explicit"/>
</element>
<command name="Point">
<input a0="h"/>
<output a0="G"/>
</command>
<element type="point" label="G">
<show object="true" label="true"/>
<objColor r="0" g="51" b="204" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-24" y="1"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<coords x="-1.0" y="-3.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="E" a1="F"/>
<output a0="i"/>
</command>
<element type="segment" label="i">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="4.0" z="20.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="E" a1="G"/>
<output a0="j"/>
</command>
<element type="segment" label="j">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.0" y="0.0" z="-2.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Line">
<input a0="F" a1="j"/>
<output a0="k"/>
</command>
<element type="line" label="k">
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.0" y="0.0" z="6.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="h" a1="k"/>
<output a0="H"/>
</command>
<element type="point" label="H">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="5" y="-1"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="6.0" y="-6.0" z="2.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="E" a1="G" a2="H" a3="F"/>
<output a0="poly2" a1="e_1" a2="g_1" a3="h_1" a4="f_1"/>
</command>
<element type="polygon" label="poly2">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="e_1">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2.0" y="0.0" z="-2.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="g_1">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="4.0" z="12.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="h_1">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.0" y="0.0" z="-6.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="f_1">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="-4.0" z="-20.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="Textpoly2" exp=""Area " + (Name[E]) + (Name[G]) + (Name[H]) + (Name[F]) + " = " + poly2" />
<element type="text" label="Textpoly2">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" size="4" style="1"/>
<absoluteScreenLocation x="476" y="362"/>
</element>
</construction>
</geogebra>