Trapezoid and Its Geometric Properties
2014-11-11 11:52
This GeoGebra applet dynamically demonstrates the fundamental geometric properties of a trapezoid. By constructing its vertices, sides, and diagonals, users can intuitively observe the relationships between interior angles, side lengths, and the properties of parallel sides. Combining angle measurement and segment analysis, this model helps learners deeply understand the definition of a trapezoid
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.27.0" id="ae1ce741-1a35-4e95-a2f4-cb9fcf8971f3" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1600" height="860" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.125" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1" size="400" window="697,247,692,546" />
<view id="4" toolbar="0 59 || 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 | 1011 || 1008 1009 || 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="true" location="1" size="1368" window="100,100,600,400" />
<view id="2" visible="true" inframe="false" stylebar="false" location="3" size="200" 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" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<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 , 22 , 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" size="300" 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 73 , 14 66 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="true" cmd="true" top="false" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
<size width="1368" height="666"/>
<coordSystem xZero="295.17152499336487" yZero="432.0721279610788" scale="74.35679840963881" yscale="74.35679840963881"/>
<evSettings axes="false" grid="true" gridIsBold="false" pointCapturing="2" 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"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0"/>
<coordStyle val="0"/>
<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="梯形" author="" date="10 12 月 2013">
<element type="point" label="A_1">
<show object="false" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="51" y="-1"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="0.0" y="3.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="point" label="B_1">
<show object="false" 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="3.0" y="3.0" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Line">
<input a0="A_1" a1="B_1"/>
<output a0="a"/>
</command>
<element type="line" label="a">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="3.0" z="-9.0"/>
<lineStyle thickness="1" type="10" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<element type="point" label="C">
<show object="true" label="true"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="-3" y="-80"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="0.0" y="0.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Line">
<input a0="C" a1="a"/>
<output a0="b"/>
</command>
<element type="line" label="b">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="3.0" z="-0.0"/>
<lineStyle thickness="1" type="15" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Point">
<input a0="a"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="true"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-1.0" y="3.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="a"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="true" label="true"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="4.0" y="3.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="A" a1="B"/>
<output a0="c"/>
</command>
<element type="segment" label="c">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="5.0" z="-15.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A" a1="C"/>
<output a0="d"/>
</command>
<element type="segment" label="d">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.0" y="1.0" z="-0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Point">
<input a0="b"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="true" label="true"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="3.0" y="0.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="B" a1="D"/>
<output a0="e"/>
</command>
<element type="segment" label="e">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.0" y="-1.0" z="-9.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="D" a1="C"/>
<output a0="f"/>
</command>
<element type="segment" label="f">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="-3.0" z="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="A" a1="B" a2="D"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<angleStyle val="0"/>
<value val="1.2490457723982544"/>
<show object="true" label="false"/>
<condition showObject="α ≟ 90°"/>
<objColor r="0" g="100" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="-43" y="16"/>
<labelMode val="1"/>
<lineStyle thickness="4" type="0" typeHidden="2"/>
<arcSize val="30"/>
</element>
<command name="Angle">
<input a0="B" a1="D" a2="C"/>
<output a0="β"/>
</command>
<element type="angle" label="β">
<angleStyle val="0"/>
<value val="1.892546881191539"/>
<show object="false" label="false"/>
<condition showObject="β ≟ 90°"/>
<objColor r="0" g="100" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="-35" y="-21"/>
<labelMode val="1"/>
<lineStyle thickness="4" type="0" typeHidden="2"/>
<arcSize val="30"/>
</element>
<expression label="text1" exp=""平行四邊形!!!""/>
<element type="text" label="text1">
<show object="true" label="false"/>
<condition showObject="c ≟ f ∧ (α ≠ 90°)"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="4.0" size="36" style="0"/>
<startPoint x="5.050351796320244" y="1.1555381099250206" z="1.0"/>
</element>
<expression label="text1_1" exp=""梯形""/>
<element type="text" label="text1_1">
<show object="true" label="false"/>
<condition showObject="(c ≠ f) ∧ (c ≠ 0) ∧ (f ≠ 0) ∧ (α ≤ 180°) ∧ (β ≤ 180°)"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="4.0" size="36" style="0"/>
<startPoint x="7.037687019631587" y="2.313233265806223" z="1.0"/>
</element>
<command name="Segment">
<input a0="A" a1="B"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
<show object="false" label="false"/>
<condition showObject="c ≟ f"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="5.0" z="-15.0"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="C" a1="D"/>
<output a0="h"/>
</command>
<element type="segment" label="h">
<show object="false" label="false"/>
<condition showObject="c ≟ f"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="3.0" z="0.0"/>
<lineStyle thickness="7" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text2" exp=""直角""/>
<element type="text" label="text2">
<show object="false" label="false"/>
<condition showObject="(α ≟ 90° ∧ β ≟ 90° ∨ η ≟ 90° ∧ θ ≟ 90°) ∧ (c ≠ f)"/>
<objColor r="127" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="4.0" size="36" style="0"/>
<startPoint x="5.035872245137572" y="2.455091129671877" z="1.0"/>
</element>
<command name="Angle">
<input a0="D" a1="C" a2="A"/>
<output a0="γ"/>
</command>
<element type="angle" label="γ">
<angleStyle val="0"/>
<value val="1.892546881191539"/>
<show object="true" label="false"/>
<condition showObject="γ ≟ δ ∧ ε ≟ ζ"/>
<objColor r="0" g="0" b="255" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="22" y="-15"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
</element>
<command name="Angle">
<input a0="B" a1="D" a2="C"/>
<output a0="δ"/>
</command>
<element type="angle" label="δ">
<angleStyle val="0"/>
<value val="1.892546881191539"/>
<show object="true" label="false"/>
<condition showObject="γ ≟ δ ∧ ε ≟ ζ"/>
<objColor r="0" g="0" b="255" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="-28" y="-21"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
</element>
<command name="Angle">
<input a0="C" a1="A" a2="B"/>
<output a0="ε"/>
</command>
<element type="angle" label="ε">
<angleStyle val="0"/>
<value val="1.2490457723982544"/>
<show object="true" label="false"/>
<condition showObject="γ ≟ δ ∧ ε ≟ ζ"/>
<objColor r="255" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="-79" y="0"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
</element>
<command name="Angle">
<input a0="A" a1="B" a2="D"/>
<output a0="ζ"/>
</command>
<element type="angle" label="ζ">
<angleStyle val="0"/>
<value val="1.2490457723982544"/>
<show object="true" label="false"/>
<condition showObject="γ ≟ δ ∧ ε ≟ ζ"/>
<objColor r="255" g="0" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="1" y="-1"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
</element>
<expression label="text3" exp=""等腰""/>
<element type="text" label="text3">
<show object="false" label="false"/>
<condition showObject="ε ≟ ζ ∧ γ ≟ δ ∧ (γ ≠ 90°)"/>
<objColor r="127" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<font serif="false" sizeM="4.0" size="36" style="0"/>
<startPoint x="5.0297350381530315" y="2.461607299239769" z="1.0"/>
</element>
<command name="Angle">
<input a0="C" a1="A" a2="B"/>
<output a0="η"/>
</command>
<element type="angle" label="η">
<angleStyle val="0"/>
<value val="1.2490457723982544"/>
<show object="true" label="false"/>
<condition showObject="η ≟ 90° ∧ θ ≟ 90°"/>
<objColor r="0" g="100" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="6" y="18"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
</element>
<command name="Angle">
<input a0="D" a1="C" a2="A"/>
<output a0="θ"/>
</command>
<element type="angle" label="θ">
<angleStyle val="0"/>
<value val="1.892546881191539"/>
<show object="true" label="false"/>
<condition showObject="η ≟ 90° ∧ θ ≟ 90°"/>
<objColor r="0" g="100" b="0" alpha="0.1"/>
<layer val="0"/>
<labelOffset x="14" y="-23"/>
<labelMode val="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
</element>
<command name="Line">
<input a0="A" a1="B"/>
<output a0="i"/>
</command>
<element type="line" label="i">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="5.0" z="-15.0"/>
<lineStyle thickness="2" type="15" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Line">
<input a0="C" a1="D"/>
<output a0="j"/>
</command>
<element type="line" label="j">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="3.0" z="0.0"/>
<lineStyle thickness="2" type="15" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<expression label="text4" exp=""三角形""/>
<element type="text" label="text4">
<show object="true" label="false"/>
<condition showObject="c ≟ 0 ∧ (f ≠ 0) ∨ f ≟ 0 ∧ (c ≠ 0)"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="4.0" size="36" style="0"/>
<startPoint x="6.035569582577504" y="2.463834451471182" z="1.0"/>
</element>
<expression label="text5" exp=""正方形""/>
<element type="text" label="text5">
<show object="true" label="false"/>
<condition showObject="c ≟ d ≟ e ≟ f"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="4.0" size="36" style="0"/>
<startPoint x="6.0297526858230315" y="2.4547144912742898" z="1.0"/>
</element>
<expression label="text6" exp=""長方形""/>
<element type="text" label="text6">
<show object="true" label="false"/>
<condition showObject="c ≟ f ∧ d ≟ e ∧ (e ≠ f) ∧ β ≟ °"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="4.0" size="36" style="0"/>
<startPoint x="6.027035683948635" y="2.4547144912742898" z="1.0"/>
</element>
<command name="Line">
<input a0="A" a1="C"/>
<output a0="k"/>
</command>
<element type="line" label="k">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.0" y="1.0" z="-0.0"/>
<lineStyle thickness="2" type="15" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Line">
<input a0="B" a1="D"/>
<output a0="l"/>
</command>
<element type="line" label="l">
<show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.0" y="-1.0" z="-9.0"/>
<lineStyle thickness="2" type="15" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
</construction>
</geogebra>