等高三角形的面积关系
2016-11-08 16:16
本范例演示了等高三角形的几何性质及其面积关系。通过构造具有相同高度的三角形，利用多边形、交点和距离等工具，直观展示底边长度与三角形面积之间的正比关系。用户可以观察并计算不同等高三角形的面积比，深入理解平面几何中三角形面积公式的实际应用。
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.288.0" id="b04e03bb-de28-4cc7-9ced-f56d993fc8b9" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1456" height="876" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.2850274725274725" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="370,155,700,550" />
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 | 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="false" location="1,1,1" size="500" 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="false" 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="false" location="1,3" size="300" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="false" location="1" size="1009" window="100,100,600,400" />
<view id="2" visible="true" inframe="false" stylebar="false" location="3" size="415" window="100,100,250,400" />
<view id="16" visible="false" inframe="false" stylebar="false" 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" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="70" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 39 73 62 | 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" position="1" help="false" />
<input show="true" cmd="true" top="algebra" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="24"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1009" height="632"/>
<coordSystem xZero="369.99999999999994" yZero="184.00000000000009" scale="50.00000000000003" yscale="50.000000000000014"/>
<evSettings axes="false" grid="true" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" 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="0"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<algebraView>
<collapsed val="2,3,4,6"/>
</algebraView>
<kernel>
<uses3D val="true"/>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="point" label="A_1">
<show object="true" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-5.0" y="-1.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="B">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="5.38" y="2.08" z="1.0"/>
<pointSize val="5"/>
<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"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="0.0" y="-2.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="A_1" a1="B" a2="C"/>
<output a0="A1" a1="c" a2="a" a3="e"/>
</command>
<element type="polygon" label="A1">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="true"/>
<objColor r="153" g="51" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="c">
<show object="true" label="true"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-3.08" y="10.379999999999999" z="-5.0200000000000005"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="a">
<show object="true" label="true"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="4.08" y="-5.38" z="-10.76"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="e">
<show object="true" label="true"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-1.0" y="-5.0" z="-10.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="point" label="D">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="1.7" y="-7.3" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="D" a1="C" a2="A_1"/>
<output a0="A2" a1="b" a2="d" a3="c_1"/>
</command>
<element type="polygon" label="A2">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="true"/>
<objColor r="153" g="51" b="0" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="b">
<show object="true" label="true"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-5.3" y="-1.7" z="-3.4"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="d">
<show object="true" label="true"/>
<objColor r="153" g="51" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-1.0" y="-5.0" z="-10.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<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"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="6.3" y="6.7" z="38.2"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="文本多边形2" exp="" " + (Name[D]) + (Name[C]) + (Name[A_1]) + " 的面积 = " + A2" />
<element type="text" label="文本多边形2">
<show object="true" label="true" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint exp="点多边形2"/>
</element>
<command name="PointIn">
<input a0="A2"/>
<output a0="点多边形2"/>
</command>
<element type="point" label="点多边形2">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<coords x="-1.5119999999999985" y="-3.6320000000000023" z="1.0"/>
<coordStyle style="cartesian3d"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="文本多边形1" exp="" " + (Name[A_1]) + (Name[B]) + (Name[C]) + " 的面积 = " + A1" />
<element type="text" label="文本多边形1">
<show object="true" label="true" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint exp="点多边形1"/>
</element>
<command name="PointIn">
<input a0="A1"/>
<output a0="点多边形1"/>
</command>
<element type="point" label="点多边形1">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<auxiliary val="true"/>
<coords x="-1.3296000000000046" y="-0.5850285714285736" z="1.0"/>
<coordStyle style="cartesian3d"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Line">
<input a0="A_1" a1="C"/>
<output a0="f"/>
</command>
<element type="line" label="f">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.0" y="5.0" z="10.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Line">
<input a0="D" a1="B"/>
<output a0="g"/>
</command>
<element type="line" label="g">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-9.379999999999999" y="3.6799999999999997" z="42.81"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="g" a1="f"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="true" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="177.25" y="-136.60999999999999" z="50.57999999999999"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="Arearatio" exp="x = A1 / A2" type="line" />
<element type="line" label="Arearatio">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.0" y="0.0" z="-1.0395161290322579"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<expression label="e_1" exp="a / a" />
<element type="numeric" label="e_1">
<value val="1.0"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="a_1" exp="a / b" />
<element type="numeric" label="a_1">
<value val="1.2131036230504926"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="Ratio2" exp="y = a / b" type="line" />
<element type="line" label="Ratio2">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="1.0" z="-1.2131036230504926"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="explicit"/>
</element>
<command name="Distance">
<input a0="B" a1="E"/>
<output a0="m"/>
</command>
<element type="numeric" label="m">
<value val="5.135638394164645"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="p" exp="Name[B] + (Name[E]) + " = " + m" />
<element type="text" label="p">
<show object="false" label="true" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint exp="Midpoint[B, E]"/>
</element>
<command name="Distance">
<input a0="E" a1="g"/>
<output a0="距离Eg"/>
</command>
<element type="numeric" label="距离Eg">
<value val="0.0"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Distance">
<input a0="D" a1="E"/>
<output a0="n"/>
</command>
<element type="numeric" label="n">
<value val="4.940412419522233"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="q" exp="Name[D] + (Name[E]) + " = " + n" />
<element type="text" label="q">
<show object="false" label="true" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint exp="Midpoint[D, E]"/>
</element>
<expression label="Ratio1" exp="z = m / n" type="plane" />
<element type="plane3d" label="Ratio1">
<show object="true" label="true"/>
<objColor r="99" g="219" b="219" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="0.0" z="1.0" w="-1.0395161290322583"/>
<fading val="0.1"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<expression label="文本1" exp=""x =Area ratio = A1: A2""/>
<element type="text" label="文本1">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint x="-6.719999999999996" y="2.8400000000000003" z="1.0"/>
</element>
<expression label="文本2" exp=""z=Ratio1=BE/DE""/>
<element type="text" label="文本2">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint x="6.419999999999992" y="1.6000000000000005" z="1.0"/>
</element>
<expression label="文本3" exp=""y=Ratio2=BC/CD""/>
<element type="text" label="文本3">
<show object="true" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint x="6.3599999999999985" y="2.4200000000000013" z="1.0"/>
</element>
</construction>
</geogebra>