Hand-in-Hand Congruence Model and Rotation Transformation
2020-07-19 02:27
This GeoGebra applet demonstrates the classic 'Hand-in-Hand' congruence model and rotation transformation. By constructing two congruent or similar polygons sharing a common vertex, it uses rotation to illustrate corresponding relationships. This model is widely used to solve geometry problems involving equal segments and angles, serving as an essential dynamic visualization tool for understanding
In collections
Minority Pi GeoGebra Collection
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.591.0" app="classic" platform="w" id="a10c1d3b-3993-457f-9305-7be83d6f07b8" xsi:noNamespaceSchemaLocation="https://ggb123.cn/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1329" height="1060" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.801354401805869" orientation="1" />
<pane location="0" divider="0.14070729872084273" 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 , 538 | 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="1" visible="true" inframe="false" stylebar="true" location="3,1" size="870" 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="2" visible="true" inframe="false" stylebar="true" location="3,3" size="187" 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="16" visible="false" inframe="false" stylebar="true" location="3,1" size="544" 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="false" 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="4097" visible="true" inframe="false" stylebar="true" location="1" size="256" window="100,100,700,550" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="false" 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="false" cmd="true" top="true" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="18"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="870" height="1060"/>
<coordSystem xZero="171.76764812339206" yZero="368.20851752876837" scale="59.36038529829549" yscale="59.36038529829559"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<bgColor r="0" g="51" b="0"/>
<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>
<mode val="1"/>
</algebraView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<expression label="A" exp="(0, 0)" type="point"/>
<element type="point" label="A">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="0" y="0" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="B" exp="A + (4, 0)" type="point" />
<element type="point" label="B">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="4" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<element type="angle" label="α">
<angleStyle val="3"/>
<value val="1.0995574287564276"/>
<slider min="0" max="90°" absoluteScreenLocation="true" width="180" x="549" y="70" fixed="false" horizontal="true" showAlgebra="true"/>
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="255" alpha="1"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="10" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
<animation step="0.017453292519943295" speed="1" type="0" playing="false"/>
</element>
<command name="Rotate">
<input a0="A" a1="(-α)" a2="B"/>
<output a0="A'"/>
</command>
<element type="point" label="A'">
<show object="false" label="true" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.1840380010418126" y="3.564026096753471" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Rotate">
<input a0="B" a1="α" a2="A"/>
<output a0="B'"/>
</command>
<element type="point" label="B'">
<show object="false" label="true" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.8159619989581874" y="3.564026096753471" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="Line[A', B]" a1="Line[B', A]"/>
<output a0="C"/>
</command>
<element type="point" label="C">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="25.88854381999832" y="50.80912807335913" z="12.94427190999916"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="A" a1="B" a2="C"/>
<output a0="t1" a1="c" a2="a" a3="b"/>
</command>
<element type="polygon" label="t1">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="c">
<show object="true" label="false"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0" y="4" z="0"/>
<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"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-3.9252210110103003" y="-2" z="15.700884044041201"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="b">
<show object="true" label="false"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="3.9252210110103003" y="-2" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="numeric" label="d">
<value val="0.46"/>
<show object="true" label="true"/>
<objColor r="255" g="215" b="0" alpha="1"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="1" absoluteScreenLocation="true" width="200" x="556" y="127" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="0.01" speed="1" type="0" playing="false"/>
</element>
<element type="angle" label="β">
<angleStyle val="3"/>
<value val="0.5585053606381855"/>
<slider min="0" max="6.283185307179586" absoluteScreenLocation="true" width="180" x="559" y="198" fixed="false" horizontal="true" showAlgebra="true"/>
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="255" alpha="1"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="10" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
<animation step="0.017453292519943295" speed="1" type="0" playing="false"/>
</element>
<command name="Rotate">
<input a0="Polygon[Dilate[A, d, C], Dilate[B, d, C], Dilate[C, d, C]]" a1="(-β)" a2="C"/>
<output a0="t2"/>
</command>
<element type="polygon" label="t2">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<command name="Vertex">
<input a0="t2"/>
<output a0="F_1" a1="F_2" a2="F_3"/>
</command>
<element type="point" label="F_1">
<show object="false" label="true" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.2629726456867325" y="2.8815096796298247" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<element type="point" label="F_2">
<show object="false" label="true" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.823381142614556" y="1.9064582334407278" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<element type="point" label="F_3">
<show object="false" label="true" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="25.88854381999832" y="50.80912807335913" z="12.94427190999916"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Vertex">
<input a0="t2"/>
<output a0="E_1" a1="E_2" a2="E_3"/>
</command>
<element type="point" label="E_1">
<show object="false" label="true" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-12" y="-8"/>
<labelMode val="3"/>
<coords x="0.2629726456867325" y="2.8815096796298247" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="E"/>
</element>
<element type="point" label="E_2">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-7" y="30"/>
<labelMode val="3"/>
<coords x="1.823381142614556" y="1.9064582334407278" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="E"/>
</element>
<element type="point" label="E_3">
<show object="false" label="true" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="25.88854381999832" y="50.80912807335913" z="12.94427190999916"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Vertex">
<input a0="t2"/>
<output a0="D_1" a1="D_2" a2="D_3"/>
</command>
<element type="point" label="D_1">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-22" y="8"/>
<labelMode val="3"/>
<coords x="0.2629726456867325" y="2.8815096796298247" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
<caption val="D"/>
</element>
<element type="point" label="D_2">
<show object="false" label="true" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.823381142614556" y="1.9064582334407278" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<element type="point" label="D_3">
<show object="false" label="true" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="25.88854381999832" y="50.80912807335913" z="12.94427190999916"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="D_1" a1="C" a2="A"/>
<output a0="t3" a1="a_1" a2="d_1" a3="c_1"/>
</command>
<element type="polygon" label="t3">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<condition showObject="e"/>
<objColor r="255" g="0" b="0" alpha="0.5699999928474426"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="a_1">
<show object="true" label="false"/>
<condition showObject="e"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-1.0437113313804756" y="1.7370273543132675" z="-4.7307936050891195"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="d_1">
<show object="true" label="false"/>
<condition showObject="e"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="3.9252210110103003" y="-2" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="c_1">
<show object="true" label="false"/>
<condition showObject="e"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-2.8815096796298247" y="0.2629726456867325" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Polygon">
<input a0="E_2" a1="C" a2="B"/>
<output a0="t4" a1="b_1" a2="e_2" a3="c_2"/>
</command>
<element type="polygon" label="t4">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<condition showObject="e"/>
<objColor r="127" g="0" b="255" alpha="0.550000011920929"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="b_1">
<show object="true" label="false"/>
<condition showObject="e"/>
<objColor r="127" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-2.0187627775695725" y="0.17661885738544392" z="3.3442575051891685"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="e_2">
<show object="true" label="false"/>
<condition showObject="e"/>
<objColor r="127" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="3.9252210110103003" y="2" z="-15.700884044041201"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="c_2">
<show object="true" label="false"/>
<condition showObject="e"/>
<objColor r="127" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-1.9064582334407278" y="-2.176618857385444" z="7.625832933762911"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="boolean" label="e">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="585" y="271"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="全等"/>
</element>
</construction>
</geogebra>