Exploring Properties of Perpendicular Segments — Shortest Distance from a Point
2021-01-03 06:36
This GeoGebra example explores the fundamental properties of perpendicular segments through geometric construction. The applet includes points (A, B, C, D, E, etc.), segments, polygons, and arcs, along with interactive buttons and text annotations. It demonstrates the theorem that the perpendicular segment from a point to a line is the shortest among all segments connecting the point to the line.
In collections
Points, Lines, and Angles
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.620.0" app="classic" platform="offline" id="BCED9897-957B-414F-89D0-8DA259ED2E7F" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="800" height="572" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.125" orientation="1" />
</panes>
<views>
<view id="16" visible="false" inframe="false" stylebar="true" location="1,3,1" size="386" window="50,50,500,500" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 || 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 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<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" size="0" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="800" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3" size="320" tab="ALGEBRA" window="100,100,600,400" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="22" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="false" stylebar="false" location="1" size="480" 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="false" inframe="false" stylebar="true" location="1" size="243" window="100,100,700,550" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="900" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 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="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="16"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="800" height="476"/>
<coordSystem xZero="465.8955995352888" yZero="307.72363031533473" scale="35.74445187650398" yscale="35.74445187650373"/>
<evSettings axes="false" 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="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"/>
<collapsed val="0"/>
</algebraView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="3"/>
<angleUnit val="degree"/>
<algebraStyle val="3" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="探究垂线段的性质" author="" date="">
<expression label="A" exp="(-5, -5)" type="point"/>
<element type="point" label="A">
<show object="false" label="true"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-5" y="-5" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="B" exp="A + (5, 0)" type="point" />
<element type="point" label="B">
<show object="false" label="true"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="-5" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="C" exp="A + (10, 0)" type="point" />
<element type="point" label="C">
<show object="false" label="true"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="5" y="-5" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="D" exp="A + (10, 10)" type="point" />
<element type="point" label="D">
<show object="false" label="true"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="5" y="5" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="E" exp="A + (5, 10)" type="point" />
<element type="point" label="E">
<show object="false" label="true"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="5" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="l2" exp="{Polygon[E, B, C, D]}" />
<element type="list" label="l2">
<show object="true" label="false" ev="4"/>
<objColor r="102" g="102" b="102" alpha="0.1"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<command name="Arc">
<input a0="Circle[(0, 0), (sqrt(2) * 5)]" a1="A" a2="C"/>
<output a0="c"/>
</command>
<element type="conicpart" label="c">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<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="-50.00000000000001" A3="0" A4="0" A5="0"/>
<eqnStyle style="specific"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="G"/>
</command>
<element type="point" label="G">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="10" type="3" playing="false"/>
<coords x="-5" y="-5.000000000000001" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="H'" exp="G + (0, 10)" type="point" />
<element type="point" label="H'">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-5" y="4.999999999999999" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<expression label="F" exp="A + (0, 10)" type="point" />
<element type="point" label="F">
<show object="false" label="true"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-5" y="5" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="button" label="button1">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="138" y="115"/>
<labelMode val="3"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="P'=CopyFreeObject[P]
StartAnimation[G]
j=Segment[P',Q]
SetColor[j,blue]
SetLineThickness[j,8]
SetLineStyle[j,1]
SetFixed[P,true]
SetValue[t_1,true]"/>
<font serif="false" sizeM="2" size="16" style="0"/>
<caption val="开始探究"/>
</element>
<command name="Segment">
<input a0="B" a1="E"/>
<output a0="f"/>
</command>
<element type="segment" label="f">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelOffset x="0" y="-22"/>
<labelMode val="3"/>
<selectionAllowed val="false"/>
<coords x="-10" y="0" z="0"/>
<lineStyle thickness="12" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="$\huge l$"/>
</element>
<command name="Point">
<input a0="f"/>
<output a0="Q"/>
</command>
<element type="point" label="Q">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="1"/>
<labelOffset x="4" y="16"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="0" y="2.6220469302242417" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<caption val="拖动点Q"/>
</element>
<element type="button" label="button2">
<show object="true" label="true"/>
<condition showObject="t_1"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="138" y="162"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="H=Midpoint[P,P']
SetColor[H,red]
m=Segment[P,H]
SetColor[m,red]
SetLineThickness[m,10]
n=Segment[P',H]
SetColor[n,red]
SetLineThickness[n,10]
SetLineStyle[n,1]
SetValue[t_2,true]

"/>
<font serif="false" sizeM="2" size="16" style="0"/>
<caption val="基本事实"/>
</element>
<element type="button" label="button3">
<show object="true" label="true"/>
<condition showObject="t_2"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="139" y="212"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="SetValue[G,A]
Delete[P']
SetFixed[P,false]
SetValue[t_1,false]
SetValue[t_2,false]
"/>
<font serif="false" sizeM="2" size="16" style="0"/>
<caption val="复位"/>
</element>
<expression label="l1" exp="{Polygon[A, B, E, F]}" />
<element type="list" label="l1">
<show object="true" label="false" ev="4"/>
<objColor r="102" g="102" b="102" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="3" type="15" typeHidden="1" opacity="51"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<command name="Segment">
<input a0="PointIn[Polygon[E, H', G, B]]" a1="(-2.9120601115768174, -1.889810189278843)"/>
<output a0="g_{2}"/>
</command>
<element type="segment" label="g_{2}">
<show object="false" label="false"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.889810189278843" y="-0.41206011157681743" z="4.724525473197108"/>
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Polygon">
<input a0="E" a1="H'" a2="G" a3="B"/>
<output a0="q1" a1="e" a2="h" a3="g" a4="b"/>
</command>
<element type="polygon" label="q1">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="102" g="102" b="102" alpha="0.1"/>
<layer val="1"/>
<labelOffset x="-53" y="60"/>
<labelMode val="3"/>
<selectionAllowed val="false"/>
</element>
<element type="segment" label="e">
<show object="false" label="false" ev="4"/>
<objColor r="102" g="102" b="102" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="8.881784197001252e-16" y="-5" z="25"/>
<lineStyle thickness="1" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="h">
<show object="false" label="false" ev="4"/>
<objColor r="102" g="102" b="102" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="10" y="0" z="50"/>
<lineStyle thickness="1" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="g">
<show object="false" label="false" ev="4"/>
<objColor r="102" g="102" b="102" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="-8.881784197001252e-16" y="5" z="25"/>
<lineStyle thickness="1" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="b">
<show object="false" label="false" ev="4"/>
<objColor r="102" g="102" b="102" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="-10" y="0" z="0"/>
<lineStyle thickness="10" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Point">
<input a0="Segment[G, H']"/>
<output a0="P"/>
</command>
<element type="point" label="P">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="1"/>
<labelOffset x="0" y="34"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-5" y="-2.4136996136560587" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="P" a1="Q"/>
<output a0="i"/>
</command>
<element type="segment" label="i">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-5.0357465438803" y="5" z="-13.110234651121209"/>
<lineStyle thickness="8" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="boolean" label="t_1">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<element type="boolean" label="t_2">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<expression label="text1" exp=""
PQ=" + (LaTeX[i]) + """ />
<element type="text" label="text1">
<show object="true" label="false" ev="8"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="7" style="0"/>
<startPoint x="0.98794" y="-3.96981" z="1"/>
</element>
<expression label="text2" exp=""
探究垂线段的性质""/>
<element type="text" label="text2">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2" size="16" style="0"/>
<absoluteScreenLocation x="45" y="14"/>
</element>
<command name="Segment">
<input a0="(0, 6)" a1="(0, -6)"/>
<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="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="12" y="0" z="0"/>
<lineStyle thickness="12" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
</construction>
</geogebra>