Area of a Parallelogram: Geometric Construction and Demonstration
2012-05-01 15:18
This GeoGebra applet dynamically demonstrates the calculation principle of the area of a parallelogram. By constructing segments, orthogonal lines, and polygons, users can intuitively observe the relationship between the base and the height. The model utilizes geometric construction tools, likely employing the cut-and-paste method to transform the parallelogram into a rectangle of equal area, ther
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="4.0" id="e5bfbd69-cf5f-46d5-bbe2-e6bbb8cdcfd4" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="844" height="586" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.18838862559241706" 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="300" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="false" location="1" size="842" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="159" window="100,100,250,400" />
<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="842" height="420"/>
<coordSystem xZero="51.00000000000002" yZero="526.0000000000002" scale="50.00000000000002" yscale="50.00000000000003"/>
<evSettings axes="true" grid="false" 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="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" 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="" author="" date="">
<element type="point" label="A">
<show object="true" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="7.14" y="7.02" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="point" label="B">
<show object="false" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-20" y="-2"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="3.54" y="3.02" 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="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="13.0" y="7.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Line">
<input a0="A" a1="C"/>
<output a0="a"/>
</command>
<element type="line" label="a">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.019999999999999574" y="5.86" z="-41.279999999999994"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Line">
<input a0="B" a1="a"/>
<output a0="b"/>
</command>
<element type="line" label="b">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.019999999999999574" y="5.86" z="-17.768"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Segment">
<input a0="A" a1="B"/>
<output a0="c"/>
</command>
<element type="segment" label="c">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.9999999999999996" y="-3.5999999999999996" z="-3.2880000000000003"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Line">
<input a0="C" a1="c"/>
<output a0="d"/>
</command>
<element type="line" label="d">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.9999999999999996" y="-3.5999999999999996" z="-26.799999999999997"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="b" a1="d"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="true" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="2" y="20"/>
<labelMode val="0"/>
<coords x="221.0128" y="70.536" z="23.511999999999997"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Length">
<input a0="Segment[B, D]"/>
<output a0="e"/>
</command>
<element type="numeric" label="e">
<value val="5.860034129593445"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<element type="numeric" label="f">
<value val="0.01"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0.01" max="e" absoluteScreenLocation="true" width="100.0" x="521.0" y="46.0" fixed="true" horizontal="true"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
</element>
<command name="Circle">
<input a0="B" a1="f"/>
<output a0="g"/>
</command>
<element type="conic" label="g">
<show object="false" label="false"/>
<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="21.6519" A3="0.0" A4="-3.54" A5="-3.02"/>
<eqnStyle style="specific"/>
</element>
<command name="Intersect">
<input a0="g" a1="b" a2="2"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="true" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-16" y="2"/>
<labelMode val="0"/>
<coords x="3.549999941758918" y="3.019965870505943" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Line">
<input a0="E" a1="c"/>
<output a0="h"/>
</command>
<element type="line" label="h">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.9999999999999996" y="-3.5999999999999996" z="-3.328122633214278"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="a" a1="h"/>
<output a0="F"/>
</command>
<element type="point" label="F">
<show object="false" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="168.11079863063566" y="165.05343754733568" z="23.511999999999997"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="OrthogonalLine">
<input a0="F" a1="b"/>
<output a0="i"/>
</command>
<element type="line" label="i">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-137.78032" y="0.4702399999999899" z="981.8282112245784"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="b" a1="i"/>
<output a0="G"/>
</command>
<element type="point" label="G">
<show object="false" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="5761.86854209603" y="2428.444161535509" z="807.40208"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="F" a1="E" a2="G"/>
<output a0="poly1" a1="g_1" a2="f_1" a3="e_1"/>
</command>
<element type="polygon" label="poly1">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="51" g="51" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="g_1">
<show object="true" label="false"/>
<objColor r="51" g="51" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.9999999999999982" y="-3.6" z="-3.328122633214271"/>
<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="51" g="51" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.012239953407105375" y="3.5863063482818887" z="-10.873974606872459"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="e_1">
<show object="true" label="false"/>
<objColor r="51" g="51" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-4.012239953407104" y="0.013693651718111433" z="28.59138646547986"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="OrthogonalLine">
<input a0="A" a1="b"/>
<output a0="j"/>
</command>
<element type="line" label="j">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-5.86" y="0.019999999999999574" z="41.7"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="b" a1="j"/>
<output a0="H"/>
</command>
<element type="point" label="H">
<show object="true" label="true"/>
<objColor r="64" g="64" b="64" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="244.71736" y="103.28648000000003" z="34.34"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="A" a1="H" a2="D" a3="C"/>
<output a0="poly2" a1="a_1" a2="h_1" a3="d_1" a4="c_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="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="a_1">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="4.0122399534071045" y="-0.013693651718113209" z="-28.55126383226557"/>
<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="0.007760046592894643" y="2.273693651718114" z="-6.894025393127553"/>
<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="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-3.9999999999999996" y="3.5999999999999996" z="26.79999999999999"/>
<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="false"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.019999999999999574" y="-5.86" z="41.279999999999994"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text1" exp=""平行四邊形面積＝底 x 高""/>
<element type="text" label="text1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" size="16" style="0"/>
<startPoint x="0.28000000000000025" y="9.08" z="1.0"/>
</element>
<expression label="text2" exp=""用滑鼠移動f,調整藍色區塊位置""/>
<element type="text" label="text2">
<show object="true" label="true"/>
<objColor r="255" g="0" b="102" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" size="16" style="0"/>
<startPoint x="7.0600000000000005" y="8.379999999999999" z="1.0"/>
</element>
</construction>
</geogebra>