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(:title Triangles - Medians and Right Triangles:) >>&<< [table border=1 width=98% cellpadding=3] [row] []>>&<<(:showhide init=hide div=div8 lshow="+" lhide="-":) Worksheet Materials (Handout & Teacher Page) (:div8 id=div8 :) (:div style="margin-left:40px":) [table border=1 cellpadding=5] [row] [c]Attach:Activity/tri_median_right_hand.gif [c valign=middle style="background-color:#eeeeff; font-face:Comic Sans Ms; font-size:1.2em"]%exa% 2-page printable %newwin%[[Attach:Activity/tri_median_right.pdf|pdf]]  or  %newwin%[[Attach:Activity/tri_median_right.doc|doc]] %dkr s8%(opens in new window)%% [tableend] (:divend:) (:div8end :) >>-<< (:showhide init=hide div=div7 lshow="+" lhide="-":) Materials for Offline Use (:div7 id=div7 :) (:div style="margin-left:40px":) Zip for offline use: [[http://mathcasts.org/mtwiki/uploads/Activity/tri_median_right.zip|tri_median_right.zip]] (includes handout, teachers page, 2 ggb interactivities). Requires freeware [[http://www.geogebra.org|GeoGebra]] and [[http://www.java.com/en/download/index.jsp|sunJava player]]. (:divend:) (:div7end :) >>-<< (:showhide init=hide div=div9 lshow="+" lhide="-":) Metadata%% >>&<< (:div9 id=div9 :) [table class=column border=1 width=98% cellpadding=3] [row] [ valign=middle]Brief []User interacts with construction and notices that the triangle constructed under the given conditions "appears" to be a right triangle. He then uses standard 6th grade geometric facts to "prove" that the angle is in fact 90°. Finally, in a second interactivity the user constructs the triangle. [row] [ valign=middle]Pre-knowledge []The sum of the angles of a triangle is 180°, the sum of supplementary angles is 180° the angles at the base of an isosceles triangle are congruent (equal size). [row] []Goal []Understanding triangles, angles and logical conclusions. [row] []Grade []6-9 (6th grade, 7th grade, pre-algebra, geometry) [row] []Strand []Measurement and Geometry, Geometry [row] []Standards []%popwin%[[standards/CA_6MG-2-2?action=popopen|CA 6.MG.2.2]], %popwin%[[standards/CA_GE-13?action=popopen|CA Geometry 13.0]], %popwin%[[standards/ACTpf24?action=popopen|ACT PF 24-27]] [row] []Keywords []triangles, medians, right-triangles, construction, interactivity, geogebra [row] []Comments []Suitable for 6th-grade on up. Can be connected to [[GlossaryT/ThalesTheorem|Thale's Theorem]] on Circles, Diameters and Right-Triangles where the "longest side" is the diameter and the median is any radius. [row] []Source []Linda Fahlberg-Stojanovska (no copyright) [row] []Cost []Activity and software is free to use [row] [ valign=middle]Download []Zip for offline use: [[http://mathcasts.org/mtwiki/uploads/Activity/tri_median_right.zip|tri_median_right.zip]] (includes handout, teachers page, 2 ggb interactivities) Requires freeware [[http://www.geogebra.org|GeoGebra]] for offline use. [row] []Type []Java Applet so requires free [[http://www.java.com/en/download/index.jsp|sunJava player]] [tableend] (:div9end :) >>&<< [tableend] >>-<< [table border=1 width=98% cellpadding=3] [row] []>>&<<%navy c12% Online Activities >>-<< >>&<< [table border=1 width=650px bgcolor=#ffffee] [row] [l]%reg c11%Theorem: %%Let {$\overline{CM}\,\, $}be the %popwin%[[def/median?action=popopen|median]] to the longest side {$ \overline{AB}\,\, $}of triangle {$ \Delta ABC $}. >>&<< ->        %red%If {$\overline{CM}\,\, $} is exactly half the length {$ \overline{AB}\,\, $}%% then %ref%{$ \Delta ABC \,\,$}is a right-triangle.%% [tableend] >>&<< (:div12 style="margin-left:20px":) (:showhide init=show div=div11 lshow="+" lhide="-":) %pra c11%Interactivity 1:%% Medians and Right Triangles %s9%(:showhide init=hide div=div17 lshow="+" lhide="-":) Directions for interactivity%% (:div11 id=div11 :) (:div17 id=div17 style="font-size:.9em":) *Click and drag the %red%slider a%% to change the length of the side {$ \overline{AC} $} *Click and drag the %ref%slider m%% to change the length of the median (and the hypotenuse {$ \overline{AB} $}). *Click and drag point C to %pra%move%% and point A to %reg%rotate%%. *Click the checkboxes as desired. (:div17end :) >>&<< >>c11<< %exa%Notice that angle %pra%{$ ><< >>-<< (:applet code="geogebra.GeoGebraApplet" codebase="Java/GeoGebra/" archive="http://www.geogebra.org/webstart/geogebra.jar" width="800" height="350" filename="http://mathcasts.org/gg/student/xtra/tri_median_rightA.ggb" showResetIcon="true" showToolBar="false" showMenuBar="false":) >>&<< (:div12end:) (:div22 style="margin-left:60px":) (:showhide init=hide div=div21 lshow="+" lhide="-":) Questions to think about? (:div21 id=div21 :) *Does triangle ABC satisfy the "hypothesis" of the theorem (no matter slider or point positions)? *Select checkbox: Show Angles. (:div227 style="margin-left:20px":) #How many degrees in: %red%{$ \alpha + \beta $}%% ? #What kind of triangle is %ref%{$ \Delta CAM $}%% ? #How many degrees in: %red%{$ 2 \cdot \delta + \beta $}%% ? #Why does %ref%{$ \delta =\frac{\alpha}{2} $}%% ? #What kind of triangle is %red%{$ \Delta BMC $}%% ? #Why does %ref%{$ \frac{\alpha}{2}+ \gamma = 90^\circ $}%% ? (:div227end:) >>&<< -> %exa c11%Finally, why does %pra%{$ < ACB = 90^\circ $}%% %%? (:div21end :) >>&<< (:div22end:) (:div11end :) >>-<< (:div333 style="margin-left:20px":) [table width=100% cellpadding=5] [row] [l](:showhide init=hide div=div31 lshow="+" lhide="-":) %pra c11%Interactivity 2: Do the construction!%% (:div31 id=div31 style="margin-left:20px":) 1. %sol%Look at the construction in the '''window above''' %%. >>&<< 2. Notice that ΔABC satisfies the theorem no matter slider or point positions. %ref%This is because of the construction!%% >>&<< 3. Now %pra%create this construction in the '''window below'''%%. (:div style="margin-left:20px":) (:showhide init=hide div=div34 lshow="+" lhide="-":) Help (:div34 id=div34 :) >>&<< * You need to construct the solid lines! * To construct %red%{$ \overline{CA} $}%%, click on Attach:GgbActivity/segment_len.jpg, then on point C and then type in "a" (letter a - no quotes) and hit Enter. -> %s8% (You can also construct {$ \overline{CA} $} "by hand" with a circle with center C and radius "a" and point A - any point on this circle.)%% * Click on Attach:GgbActivity/circle_rad.jpg, then on point C and type in "m". * Do the same for point A. Then find the intersection point and name it M ... (:div34end :) (:divend:) >>-<< (:applet code="geogebra.GeoGebraApplet" codebase="Java/GeoGebra/" archive="http://www.geogebra.org/webstart/geogebra.jar" width="800" height="500" filename="http://mathcasts.org/gg/student/xtra/tri_median_rightB.ggb" customToolBar="0 | 1 || 45 | 34 || 5 | 18 || 15 | 16 || 36 | 40 | 41 | 42 |" showResetIcon="true" showToolBar="true" showMenuBar="false":) (:div31end :) (:div3end:) >>&<< [tableend] (:div333end:) >>&<< [tableend] >>&<< ------ >>&<< %rel%Related themes: * [[GlossaryT/triangleMedian|Median]] * [[GlossaryT/ThalesTheorem|Thales Theorem - Circles, Diameters and Right Triangles]] >>&<< ---- >>&<< [table width=100%] [row] [][[activity/| Attach:main/tri_purple_up_a.gif ]] [[activity| Up one level]] [r](:html:) (:htmlend:) [row] [ colspan=2] (:html:)

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