动点与面积：几何变换与轨迹探究
2020-05-03 02:00
本范例演示了通过几何变换和动点运动来探究面积变化的数学问题。利用GeoGebra的旋转、射线、交点、多边形和轨迹等工具，构建了动点与固定图形之间的面积关系。通过条件语句和轨迹命令，直观展示了面积随动点位置变化的规律，并结合函数图像探讨极值问题，帮助深入理解平面几何与代数函数的综合应用。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.579.0" app="classic" platform="w" id="dfec9b99-cfb2-40aa-bea5-3ac13ff1587f" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1883" height="1561" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.17100371747211895" orientation="1" />
<pane location="1" divider="0.4965480616038237" orientation="1" />
</panes>
<views>
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 51 | 551 550 11 , 20 22 21 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="false" inframe="false" stylebar="true" location="1,1,1" size="500" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1,3" size="935" window="100,100,600,400" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="true" location="1,1" size="300" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="true" location="1,3" size="300" window="100,100,600,400" />
<view id="4097" visible="true" inframe="false" stylebar="true" location="1,1" size="608" window="100,100,700,550" />
<view id="2" visible="true" inframe="false" stylebar="true" location="3" size="322" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="128" visible="false" inframe="false" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="43" visible="false" inframe="false" stylebar="false" location="1" size="450" window="50,50,500,500" />
</views>
<toolbar show="false" items="0 39 73 | 1 501 67 , 5 19 , 72 75 76 | 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 14 68 | 30 29 54 32 31 33 | 25 17 26 60 52 61 | 40 41 42 , 27 28 35 , 6 | 100001 100002 100003 , 100012 100015 100016 , 100017 100018 | 100004 100005 100006 100013 100014 100019 100020 100021 100022 | 100011 100008 100009 100010 | 62 100007 100024 100023 100053 100054 | 100030 100031 100032 100035 100027 100041 100042 100039 100036 100037 , 100025 100034 100033 100046 100047 | 100043 100040 100028 , 100048 100050 100049 , 100052 100045 100029 100038 , | 100060 100059 100062 100061 100026 100055 , 100064 100063 | 100058 100057 100056 100051" position="1" help="false" />
<input show="false" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="16"/>
<consProtColumns col0="true" col1="true" col2="false" col3="true" col4="false" col5="true" col6="true" col7="false"/>
<consProtocol useColors="false" addIcons="false" showOnlyBreakpoints="false"/>
<consProtNavigationBar id="1" playButton="true" playDelay="2" protButton="true" consStep="21"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="934" height="1523"/>
<coordSystem xZero="555.6354282969332" yZero="422.9601272278761" scale="30.795300309259844" yscale="30.7953003092597"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="3" grid="0"/>
<labelStyle axes="1" serif="false"/>
<axis id="0" show="true" label="x" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" label="y" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<algebraView>
<mode val="1"/>
<auxiliary show="true"/>
</algebraView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="2" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="point" label="A">
<show object="true" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="2"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-16.35429507876873" y="3.387652094438088" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="B_1" exp="A + (8, 0)" type="point" />
<element type="point" label="B_1">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="2"/>
<labelMode val="0"/>
<coords x="-8.354295078768729" y="3.387652094438088" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Rotate">
<input a0="B_1" a1="30°" a2="A"/>
<output a0="C"/>
</command>
<element type="point" label="C">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="2"/>
<labelMode val="0"/>
<coords x="-9.426091848493218" y="7.387652094438088" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="C" a1="12"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="2"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-0.5713889953142299" A3="0" A4="9.426091848493218" A5="-7.387652094438088"/>
<eqnStyle style="specific"/>
</element>
<command name="Ray">
<input a0="A" a1="B_1"/>
<output a0="f"/>
</command>
<element type="ray" label="f">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="2"/>
<labelMode val="0"/>
<coords x="0" y="8" z="-27.10121675550471"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Intersect">
<input a0="c" a1="f"/>
<output a0="B" a1=""/>
</command>
<element type="point" label="B">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="2"/>
<labelMode val="0"/>
<coords x="1.8876166504915415" y="3.3876520944380886" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="A" a1="C"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="2"/>
<labelMode val="0"/>
<coords x="-3.9999999999999996" y="6.9282032302755105" z="-88.88752249881047"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="C" a1="B"/>
<output a0="h"/>
</command>
<element type="segment" label="h">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="2"/>
<labelMode val="0"/>
<coords x="3.999999999999999" y="11.31370849898476" z="-45.87737489441389"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Circle">
<input a0="A" a1="10"/>
<output a0="d"/>
</command>
<element type="conic" label="d">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="2"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="178.93915423638987" A3="0" A4="16.35429507876873" A5="-3.387652094438088"/>
<eqnStyle style="specific"/>
</element>
<command name="Intersect">
<input a0="d" a1="f"/>
<output a0="D" a1=""/>
</command>
<element type="point" label="D">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="2"/>
<labelMode val="0"/>
<coords x="-6.35429507876873" y="3.3876520944380886" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="A" a1="B"/>
<output a0="i"/>
</command>
<element type="segment" label="i">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="2"/>
<labelMode val="0"/>
<coords x="-4.440892098500626e-16" y="18.24191172926027" z="-61.797250476183294"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A" a1="D"/>
<output a0="j"/>
</command>
<element type="segment" label="j">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="2"/>
<labelMode val="0"/>
<coords x="-4.440892098500626e-16" y="10" z="-33.876520944380886"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="numeric" label="t">
<value val="2.8"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="2"/>
<labelMode val="1"/>
<slider min="0" max="10" absoluteScreenLocation="true" width="200" x="139" y="57" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
</element>
<command name="Point">
<input a0="j" a1="t / 10"/>
<output a0="Q"/>
</command>
<element type="point" label="Q">
<show object="true" label="true" ev="4"/>
<objColor r="128" g="0" b="0" alpha="0"/>
<layer val="2"/>
<labelOffset x="-13" y="31"/>
<labelMode val="0"/>
<coords x="-13.554295078768728" y="3.387652094438088" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="PolyLine">
<input a0="A" a1="C" a2="B"/>
<output a0="k"/>
</command>
<element type="polyline" label="k">
<lineStyle thickness="7" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="2"/>
<labelOffset x="4" y="-80"/>
<labelMode val="0"/>
</element>
<command name="If">
<input a0="0 < t ≤ 4" a1="Point[k, t / 8]" a2="Point[k, 0.5 + (t - 4) / 12]"/>
<output a0="P_2"/>
</command>
<element type="point" label="P_2">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="2"/>
<labelOffset x="-1" y="35"/>
<labelMode val="0"/>
<coords x="-11.504552817575872" y="6.1876520944380875" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="A" a1="P_2" a2="Q"/>
<output a0="t1" a1="e" a2="a" a3="f_1"/>
</command>
<element type="polygon" label="t1">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0.10000000149011612"/>
<layer val="2"/>
<labelMode val="0"/>
</element>
<element type="segment" label="e">
<show object="true" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="2"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-2.7999999999999994" y="4.849742261192857" z="-62.221265749167316"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="a">
<show object="true" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="2"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="2.7999999999999994" y="-2.049742261192856" z="44.89583988474067"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="f_1">
<show object="true" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="2"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0" y="-2.8000000000000007" z="9.485425864426652"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="If">
<input a0="t ≤ 4" a1="A + ((2 * t) * UnitVector[C - A])" a2="C + ((2 * (t - 4)) * UnitVector[B - C])"/>
<output a0="P_1"/>
</command>
<element type="point" label="P_1">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="2"/>
<labelOffset x="-31" y="-3"/>
<labelMode val="0"/>
<coords x="-11.504552817575874" y="6.1876520944380875" z="1"/>
<pointSize val="7"/>
<pointStyle val="2"/>
</element>
<expression label="E" exp="(t, t1)" type="point" />
<element type="point" label="E">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="2"/>
<labelOffset x="9" y="18"/>
<labelMode val="0"/>
<coords x="2.8" y="3.919999999999998" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Locus">
<input a0="E" a1="t"/>
<output a0="loc1"/>
</command>
<element type="locus" label="loc1">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="2"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<expression label="p" exp="p(x) = If[0 < x ≤ 4, (1 / 2 * x^(2)), If[4 < x ≤ 10, (-1 / 3 * x^(2)) + (10 / 3 * x)]]" type="function"/>
<element type="function" label="p">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0"/>
<layer val="2"/>
<labelMode val="0"/>
<fixed val="true"/>
<lineStyle thickness="8" type="0" typeHidden="1" opacity="178"/>
</element>
<expression label="text1" exp=""轨迹和函数图像合一！
向量平移和路径描点合一！
if指令将路段合一！！！
感谢左右修竹老师！！！！""/>
<element type="text" label="text1">
<show object="true" label="true" ev="40"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="2"/>
<labelMode val="0"/>
<font serif="false" sizeM="2" size="16" style="0"/>
<startPoint x="-15.607427869518487" y="-2.404258832632957" z="1"/>
</element>
</construction>
</geogebra>