Triangle Area Demonstration
2013-04-01 06:52
This GeoGebra applet demonstrates and calculates the area of a triangle. By constructing geometric objects such as polygons, midpoints, segments, and intersections, it visually presents the vertices, edges, and interior region of the triangle. Combined with text labels that dynamically display relevant geometric parameters and area calculation results, it helps students deeply understand the geome
In collections
Polygons: Interactive GeoGebra Geometry Book
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="4.2" version="4.2.28.0" id="2c88f266-f6c7-4fa9-894d-1efec19422ed" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1366" height="728" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.14641288433382138" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1" size="400" window="630,133,700,550" />
<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="1339" window="100,100,600,400" />
<view id="2" visible="false" 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" />
</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="1339" height="562"/>
<coordSystem xZero="201.00000000000003" yZero="417.99999999999994" scale="49.99999999999998" yscale="49.999999999999986"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" 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="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" 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="" author="" date="">
<element type="point" label="A">
<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="11.22" y="3.76" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="a" exp="x(A)" />
<element type="numeric" label="a">
<value val="11.22"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<element type="point" label="B">
<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="-1.1" y="1.56" z="1.0"/>
<pointSize val="3"/>
<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="7.08" y="-1.24" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="b" exp="y(A)" />
<element type="numeric" label="b">
<value val="3.76"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="e" exp="x(C)" />
<element type="numeric" label="e">
<value val="7.08"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="f" exp="y(C)" />
<element type="numeric" label="f">
<value val="-1.24"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="c" exp="x(B)" />
<element type="numeric" label="c">
<value val="-1.1"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<expression label="d" exp="y(B)" />
<element type="numeric" label="d">
<value val="1.56"/>
<objColor r="0" g="0" b="0" alpha="0.1"/>
</element>
<command name="Polygon">
<input a0="A" a1="B" a2="C"/>
<output a0="多边形1" a1="c_1" a2="a_1" a3="b_1"/>
</command>
<element type="polygon" label="多边形1">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="1.0"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="c_1">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.1999999999999997" y="-12.32" z="21.639200000000002"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="a_1">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.8" y="8.18" z="-9.6808"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="b_1">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-5.0" y="4.140000000000001" z="40.5336"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="文本1" exp=""(" + a + "," + b + ")"" />
<element type="text" label="文本1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.4" size="5" style="1"/>
<startPoint exp="A + (0.3, 0.3)"/>
</element>
<expression label="文本2" exp=""(" + c + "," + d + ")"" />
<element type="text" label="文本2">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.4" size="5" style="1"/>
<startPoint exp="B + (-1.3, 0.3)"/>
<labelOffset x="-21" y="24"/>
</element>
<expression label="文本3" exp=""(" + e + "," + f + ")"" />
<element type="text" label="文本3">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.4" size="5" style="1"/>
<startPoint exp="C + (0.3, -0.3)"/>
</element>
<expression label="文本4" exp=""A(a,b),B(c,d),C(e,f) △ABC面積 =\frac{１}{２}\left\vert cf+be+ad-bc-de-af \right\vert =\frac{１}{２}\left\vert\left\vert \begin{array}{} 1& a& b \\ 1 & c& d\\ 1& e& f \\ \end{array} \right\vert \right\vert""/>
<element type="text" label="文本4">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint x="-2.940000000000002" y="8.16" z="1.0"/>
</element>
<command name="Midpoint">
<input a0="A" a1="C"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="9.15" y="1.2599999999999998" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Midpoint">
<input a0="B" a1="C"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.99" y="0.16000000000000003" z="1.0"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="A" a1="E"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.5999999999999996" y="-8.23" z="-9.447199999999999"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="B" a1="D"/>
<output a0="h"/>
</command>
<element type="segment" label="h">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.30000000000000027" y="10.25" z="-15.66"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Intersect">
<input a0="h" a1="g"/>
<output a0="F"/>
</command>
<element type="point" label="F">
<show object="false" label="true"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="225.7156" y="53.54183999999999" z="39.369"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<expression label="文本5" exp=""=\frac{1}{2}\left\vert (" + (LaTeX[c]) + ")×(" + (LaTeX[f]) + " )+(" + (LaTeX[b]) + ")×(" + (LaTeX[e]) + ")+(" + (LaTeX[a]) + ")×(" + (LaTeX[d]) + ")-(" + (LaTeX[b]) + ")×(" + (LaTeX[c]) + ")-(" + (LaTeX[d]) + ")×(" + (LaTeX[e]) + ")-(" + (LaTeX[a]) + ")×(" + (LaTeX[f]) + ")\right\vert =" + (LaTeX[0.5 * abs(c * f + b * e + a * d - b * c - d * e - a * f)]) + " "" />
<element type="text" label="文本5">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint x="-2.019999999999997" y="6.159999999999999" z="1.0"/>
</element>
<expression label="文本多边形1" exp=""△ " + (Name[A]) + "" + (Name[B]) + "" + (Name[C]) + " 的面積 = " + 多边形1 + """ />
<element type="text" label="文本多边形1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="1.4" size="5" style="1"/>
<startPoint exp="F"/>
<labelOffset x="-60" y="-4"/>
</element>
</construction>
</geogebra>