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Defining Triangles Name(s): In this lesson, you'll experiment with an ordinary triangle and with special triangles that were constructed with constraints. The constraints limit what you can change in the triangie when you drag so that certain reiationships among angles and sides always hold. By observing these relationships, you will classify the triangles. 1. Open the sketch Classify Triangles.gsp. 2. Drag different vertices of each of the four triangles to observe and compare how the triangles behave. QI Which of the kiangles seems the most flexible? Explain. BD h--.\ \ t\ / yle AL___\c LI qA------------__. * \----"--: ,/ \ M \ "<-----\U\ a2 \Alhich of the triangles seems the least flexible? Explain. -: measure each I :-f le, select three l? coints, with the rei€x loUr middle Then, in r/easure menu, le=ection. :hoose Angle. I I 3. Measure the three angles in AABC. 4. Draga vertex of LABC and observe the changing angle measures. To answer the following questions, note how each angle can change from being acute to being right or obtuse. I I I Q3 How many acute angles can a triangle have? I ;;,i:?:PJi;il: qo How many obtuse angles can a triangle have? -sed to classify rc:s. These terms f> Qg LABC can be an obtuse triangle or an acute :"a- arso be used to triangle. One other triangle in the sketch can : assify triangles according to also be either acute or obtuse. Which triangle is it? their angles. I I I I I aG Which triangle is always a right triangle, no matter what you drag? -: I 07 Which triangle is always an equiangular triangle? =ect thenlin t'e p 5. Measure the lengths of the three sides of LABC. .. t,'leasure -Fisure a tensth, a seoment. I menu, I :-coselength.l z n . t oft LABC a An- and1observe r i, r 6. Drag a vertex the changing side lengths. furonng Geometry with The Geometer's Sketchpad tory2 { Key Curriculum Press Chapter 3: Triangles . 63 Defining Triangles (continued) Scalene. isosceles, l-9 and equilateral are terms used to classify triangles by relationships among their sides. Because AABC has no I I I I I I constraints, it can be any type of triangle. aB If none of the side lengths are equal, the triangle is a scnlene triangle. If two or more of the sides are equal in length, the triangle is isosceles.lf all three sides are equal in length, itis equilaternl. Which type of triangle is AABC most of the time? I I Q9 Name a triangle besides LABC that is scalene most of the time. (Measure if you like, but if you drag things around, you should be able to tell without measuring, just by looking.) Qro Which triangle (or triangles) is (are) always isosceles? a{r Which triangle is always equilateral? Q12 For items a-e below, state whether the triangle described is possible or not possible. To check whether each is possible, try manipulating the triangles in the sketch to make one that fits the description. If it's possible to make the triangle, sketch an example on a piece of paper. a. An obtuse isosceles triangie b. An acute right triangle c. An obtuse equiangular triangle d. An isosceles right triangle e. An acute scalene triangle Q{3 Write a definition for each of these six terms: scute triangle, obtuse triangle, right trinngle, scnlene triangle, isosceles triangle, and equilateral trinngle. Use a separate sheet if necessary. ln the Display menu, choose show Ail Hidden to get an idea of how the triangles in the Glassify I EXplOre MOre sketch were constructed. 64 . see if you can come up with ways to construct right triangle, an isosceles triangle, and an equilateral triangle. Describe your methods. 1. Open a new sketch and Triangles.gsp Chapter 3: Triangles a Exploring Geometry with The Geometer's Sketchpad @ 2002 Key Curriculum Press friangte Sum Name(s): two-part invesrigation. To measure an First T:^1-: conjectureabout":::'**'i?:i':::.1*'rrinvestiga*F you,rr";;t;;,k"ff l"T;jliffi :ill:,#*trq*,fl J ;1?',3;:frilfrii t*T,J:",n"" sketch and rnvestisate *,fi3j*,J5ifu/ 1 choose Construct AAB]. l-t 2' M"usure its three q! (6rsD. anere ChooseCalculate 'uuse sdlcUlate - -- / / : once on u J seiecr ooint B I ,"?#43#:i/ " ai _ What is the sum orF the i "f"j,"t.lii*il,: l= 7. Consfruct Lii.i.',-'.i F the interio r ,?t?,*jffi lransform [',Hi'?81'r,f""#ii B:Fil'#'lf ( men,, choose Rotate' angles in any fuianele? Follow these steps to investigate why,J'ff;:l";; s. r.-orsrruct Consrruct a tine through point B paralret paraltel to At. :6' Construct the midpoints of A-B and CE. _ :ilf;:;:: I " .;;";:" cnoose Triangls itx:E::i:: tz-/tEi(j 87.2 = It mZBcA= 4e'5o Calculate the sum of the angle measures. +. Draga vertex *of the *r' tl triangte and observe n*== il:;;:::: the angle sum. t{ff,$idffi If / '.".,,".1nitJ unr"iii''io i 1 calcutation, angres. I 1 9' Give the new fuiangle 'o' Y:* of AABC. ?ii,ffi ;c enter ror ro ta ti on and rota te the interior a different color. ffi"J:,T,:idpoint as a center and rotate the interior by 180" 11. Give this new tuiangle interior a different color. B 72. 3,ff ilffit f,:f,|l',iffiff* the three triangres are rerated to each *:t:,lixftI;j*:ll#;;r?f conjecfure from e1. i:f u--' '"vtd'ut now this ,Tl, j#:",*d,..ne.r,he demonsfrates your Chapter ar tri"nge, .ii Exterior Angles in a Triangle Name(s): Anexterior angle of a triangle is formed when one of the sides is extended. An exterior angle lies outside the triangle. In this investigation, you'll discover a relationship between an exterior angle'and the sum of the measures of the two remote interior angles Sketch and lnvestigate 1. Construct AABC. --) to extend side AC. 2. Construct AC Hold down the mouse button on the Segment tool and drag right to choose the Ray tool. 3. Construct point D onA-, nICAB = 45.5o nZABC = 67.2" mtBCD=112.74 outside of the triangle. Select, in order, Measure exterior angle BCD. points B, C, and D. Then, in the Measure menu, choose Angle. A C D' Measure the remote interior a ngles ZABC and ICAB. 6. Drag parts of the triangle and look for a relationship between the Choose Galculate ["#']H$%Tiil' Calculator. Click once on a measurement to enter it into a calculation. measures of the remote interior angles and the exterior angle. I f'qr I I I How are the measures of the remote interior angles related to the measure of the exterior angle? Use the Calculator to create an expression that confirms your conjechrre. I I I rt I Select the interior; then, in the Transform menu, choose Translate. marked vector. ( il I | 10. Give the new triangle interior a different color. I 11,. Construct the midpoint of BC Double-clickthelrrn a( tz' Mark the midpoint as a center for rotation and rotate the triangle interior about this point by 180". I n. choose Rotate. IJ. Give this new triangle interior a different color. point to mark it as center. Select the interior; then, in the Transfrom menu, I I a2 Explain how the two angles that fill the exterior angle are related to the remote interior angles in the triangle. Explain how this demonstrates your conjecture from Q1. Use the back of this sheet. 56 . Chapter 3: Triangles Exploring Geometry with The Geometer's Sketchpad @ 2OO2 Key Curriculum Press Triangle lnequalities t Name(s): In this investigation, you'll discover relationships among the measures of the sides and angles in a triangle. Sketch and lnvestigate -: 'reasure a side, r=.ct it; then, in the lr4easure menu, . - rloose Length. 1. Construct a triangle. m AB- = 2.7 cm 2. Measure the lengths of mBe=3.6cm mffi=3.4cm the three sides. -:ose Calculate '':rn the Measure -enu to open the 3. Calculate the sum of any m AB-+m BT = 6.31 cm two side lengths. lalculator. Click once on a 'neasurement to enter it into a calculation. 4. Drag a vertex of the triangle to try to make the sum you calculated equal to the length of the third side. Ql Is it possible for the sum of two side lengths in a triangle to be equal to the third side length? Explain. Q2 Do you think it's possible for the sum of the lengths of any two sides of a triangle to be less than the length of the third side? Explain. t Q3 Summarize your findings as a conjecture about the sum of the lengths of any two sides of a triangle. To measure an =-ole, select three [, 5. Measure IABC, ICBA, and IACB. points, with the 6. Drag the vertices of your triangle and look for relationships between .:1ex your middle Then, in =:lection. --: Measure choose menu, Angle. side lengths and angle measures. I Q4 In each area of the chart below, a longest or shortest side is given. Fill in the chart with the name of the angle with the greatest or least measure, given that longest or shortest side. longesl side AB AB shortest side longesl Largest angle? AC longesl Largest angle? BC Smallest angle? AC lo.tgest Smallest angle? BZlongest side side side side Largest angle? Smallest angle? Q5 Summarize your findings from the chart as a conjecture. ! :rnloring Geometry with The Geometer's Sketchpad 3 20O2 Key Curriculum Press Chapter 3: Triangles . 67 Triangle Congruence Name(s): If the three sides of one triangle are congruent to three sides of another triangle (SSS), must the two triangles be congruent? What if two sides and the angle between them in one triangle are congruent to two sides and the angle between them in another triangle (SAS)? Which combinations of parts guarantee congruence and which don In this activity, you investigate that question. Sketch and lnvestigate 1. Open the sketch Triangle Congluence.gsp. You'll like the one shown below, along with some text. see a figure Side-Side-Side ABDE 2. Read the text in the sketch and follow the instructions to try to make a triangle DEF that is not congruent to A,ABC. Q{ Could you form a triangle with a different size or shape given the three sides? A2 If three sides of one triangle are congruent to three sides of another (SSS), what can you say about the triangles? Write a conjecture that summarizes your findings. Click on the page tabs at the bottom of ,-). the sketch window to switch pages. 4. Q3 68 . Chapter 3: Triangles Go to the SAS page and make a triangle DEF with those given parts. Try to make a triangle that is not congruent to LABC. Experiment on each of the other pages. Of SSS, SAS, SSA, ASA, AAS, and AAA, which combinations of corresponding parts guarantee congruence in a pair of triangles? Which do not? Exploring Geometry with The Geometer's Skeichpad @ 2002 Key Curriculum Press rt Properties of lsosceles Triangles Name(s): In this activity, you'lllearn how to cor. :ruct an isosceles triangle (a triangle with at least two sides the same length). Then you'll discover properties of isosceles triangles. Sketch and lnvestigate L. Construct a circle with center A and radius point B. 2. Construct radius AB. a J. Construct radius AC.Dragpoint C to make sure the radius is attached to the circle. 4. Construct 5. Drag each vertex of your triangle to see how it behaves. BZ c Q{ Explain why the triangle is always isosceles. 6. Hide circle AB. To measure an I "'rgle, select three l? points, with the .er1ex your middle selection. Then, in :e Measure menu, I I 7. Measure the three angles in the triangle. 8. Drag the vertices of your triangle and observe the angle measures. I choose Angle. I I a2 Angle s ACB and. ABCare the base anglesof the isosceles triangle. Angle CAB is the aertex angle. What do you observe about the angle measures? Explore More 1. Investigate the converse of the conjecfure you made in Q2: Draw any triangle and measure two angles. Drag a vertex until the two angle measures are equal. Now measure the side lengths. Is the triangle isosceles? 2. Investigate the statement if a triangle is equiangular, then it is equilateral and its converse, if a triangle is equilateral, then it is equiangular. Can you construct an equiangular triangle that is not equilateral? Can you construct an equilateral triangle that is not equiangular? Write a conjecfure based on your investigation. l":ploring Geometry with The Geometer's Sketchpad g 2002 Key Curriculum Press Chapter 3: Triangles . 69 Gonstructing lsosceles Triangles Name(s): How many ways can you come up with to construct an isosceles triangle? Try methods that use just the freehand tools (those in the Toolbox) and also mEthods that use the Construct and Transform menus. Write a brief description of each construction method along with the properties of isosceles triangles that make that method work. Method 1: Properties: Method 2: Properties: Method 3: Properties: Method 4: Properties: 70 . Chapter 3: Triangles Exploring Geometry with The Geometer's Sketch@ @2002 Key Cuniculum Press Medians in a Triangle t Name(s): A median in a triangle connects a vertex with the midpoint of the opposite side. In previous investigations, you may have discovered properties of angle bisectors, perpendicular bisectors, and altitudes in a triangle. Would you care to make a guess about medians? You may see what's coming, but there are new things to discover about medians, too. Sketch and lnvestigate L. Construct triangle ABC. ' _,3u've already ::-s:ructed three 2. Construct the midpoints of the three sides. 3. Construct two of the three medians, each connecting a vertex with the midpoint of its opposite side. 4. Construct the point of intersection of the two medians. *edians, select :.. - :i them. Then, - :re Construct renu, choose 5. Construct the third median. lntersection. Q{ What do you notice about this third median? Drag a vertex of the triangle to confirm that this conjecture holds for any triangle. ! - / l-- o' The point where the medians :-: ooint to show intersect is called the centroid. :s abel. Double: ck the label to Show its label and change it change it, to Ce for centroid. -sino the Text I ,: clrEk nn.. nn I | I I -o measure the r stance between 7. Measure the distance from B to Ce and the distance from Ce to the midpoint F. 8. Drag vertices of AABC and look for a relationship between BCe and CeF. 2j7 Make a table with these two measures. :,',c points, select :-em; then, in the :'/easure menu, --:ose Distance. Select the two measurements; then, in the Graph menu, BCe 9. CeF cm 1.08 cm 1.00 cm 0.50 cm 3.24 cm 1.62 cm ?..tJlj 1"tt7 *ns t:ry1 Change the triangle and double-click the table values to add another entry. 71. Keep changing the triangle and adding entries to your table until you can see a relationship between the distances BCe and CeF. -xploring Geometry with The Geometer's Sketchpad O 2002 Key Curriculum Press Chapter 3: Triangles . 71 f,ledians in a Triangle (continued) g*ed on what you notice about the table erttries, use the Calculator to "1;;",fil".:f,J,: l-tfZ. even make an expression with the measures that n'ill remain constant ryl;ii"*liii I --;;;";;; Calculator. Click "- measurement to ":5:[$3f I I as the measures change. I I o, Write the expression you calculated in step 1'2' e3 Write a conjecture about the way the centroid divides a 13. 1.4. each median in triangle. Plot the table data on a piece of graph Paper or using the Plot points command in thebraph menu. You should get a graph with several collinear Points. Draworconstructalinethroughanytwoofthedatapointsand measure its sloPe. QaExplainthesignificanceoftheslopeofthelinethrough the data Points. Explore More tool that constructs the centroid of a triangle' Save your sketch in the Tool Folder (next to the sketchpad application for future itself on your hard drive) so that the tool will be available investigations of triangle centers' 1. Make 72 . Chapler 3: Triangles a custom Exploring Geometry with The Geometer's @2OO2 KeY Curriculum Perpendicular Bisectors in a Triangle Name(s): In this investigatiory you'rl discover properties of in a triangle. you'rl aiso rearn how to construct perpendicular bisectors a circre that passes through each vertex of a triangle. Sketch and lnvestigate l: 1. Construct triangle ABC. 2. Construct the midpoints of the sides. :ct a side and . ridpoint; then, - :he Construct 3. Construct two of the three .nenu, choose Perpendicular perpendicular bisectors in the triangle. Line. Click afthe -'=--:ction with the IL- Selection Arrow :col or with the Point toot. 4. Construct point G the point of intersection of these lines. I I i 5. Construct the third perpendicular bisector. Q{ what do you notice about this third perpendicurar bisector (not Drag a vertex of the triangle to conJirm that this conjecture :h?y")? a holds for any triangle. Drag a vertex around so that point G moves into and out of the triangle' observe the angles of the triangre as you ao tr,ir. a2 In what type of triangre is point G outside the triangre? inside the triangle? 6. Q3 := ect point G and a .ertex; then, in the Measure menu, :.oose Distance. ::peat for the other two vertices. Drag a vertex until point G falls on a side of the triangle. What kind of triangle is this? Where exactly does point G lie? Measure the distances from point G to each of the three vertices. 8. Drag a vertex of the triangre and observe the distances. Q4 The point of intersection of the three perpendicurar bisectors is carled the circumcenter of the hiangle. what do you notice about the distances from the circum.".-.,t", to the tFLree vertices of tn" ftiangle? 7. D -xploring Geometry with The Geometer,s SketchpaJ ! 2002Key Curriculum press Chapter 3: Triangles . 73 Perpendicular Bisectors in a Triangle (continued) 9. Construct a circle with center G and radius endpoint A. This is the circumscribed circle of LABC. Explore More 1. Make tool that constructs the circumcenter of a triangle. Save your sketch in the Tool Folder (next to the Sketchpad application itself on your hard drive) so that the tool will be available for future investigations of triangle centers. 2. a custom Explain why the circumcenter is the center of the circumscribed circle. Hint: Recall that any point on a segment's perpendicular bisector is equidistant from the endpoints of the segment. Why would the circumcenter be equidistant from the three vertices of the triangle? if you can circumscribe other shapes besides triangles. Describe what you try and include below any additional conjectures you come up with. J. See 74 . Chapter 3: Triangles Exploring Geometry with The Geometer's Sketchpad @ 2002 Key Curriculum Press Altitudes in a Triangle Name(s): In this activity, you'll discover some properties of altitudes in a triangle. An nltitude is a perpendicular segment from a vertex of a triangle to the opposite side (or to a line containing the side). The side where the altitude ends is thebase for that altitude, and the length of the altitude is the height of the triangle from that base. Because a triangle has three sides, it also has three altitudes. You'll construct one altitude and make a custom tool for the construction. Then you'll use your tool to construct the other two, Sketch and lnvestigate 1. Construct triangle ABC. 2. Construct a line perpendicular to At through point Ql B. As long as your triangle is acute, this perpendicular line should intersect a side of the triangle. Drag point B so that the line falls outside the triangle. Now what kind of triangle is it? a J. With the perpendicular line outside the triangle, use a line to extend side AC so that it intersects the perpendicular line. 4. Construct point D, the point of intersection of the extended side and the perpendicular line. 5. Hide the iines. 6. Construct nD. Segment BD is an altitude. Steps 3 and 44 Steps 5 and 6 7. Drag vertices of the triangle and observe how your altitude behaves. Q2 Where is your altitude when AiltilMrr,l Geometry with The Geometer's Sketchpad F tr1li;: <eV Curriculum Press lAis a right angle? Chapter 3: Triangles . 75 Altitudes in a Triangle (continued) Select evervthino in | vori .r"i.'n. iii"n, press the Gustom tools icon (the bottom tool in the Toolbox) and choose 8. Drag your kiangle so that it is acute again (with the altitude falling inside the triangle). p. Make a custom tool for this construction. Name the tool "Altitude." I I I f> Greate New Tool l from the menu il;i;;#;i; I I Starting with three points (the tool's "givens"), the Altitude tool will construct a triangle and an altitude from one of the vertices. To use the tool, click on the Custom tools icon to select the most recently created tool, then click on the three triangle vertices. 10 Use your custom tool on the triangle's vertices to construct a second altitude. Don't worry if you accidentally construct the altitude that already exists. Just use the tool on the vertices again in a different order until you get another altitude. 11. Use your Altitude tool to construct the third altitude in the triangle. 12. Drag the triangle and observe how the three altitudes behave. Q3 What do you notice about the three altitudes when the triangle is acute? Step 11 Qa What do you notice about the altitudes when the triangle is obtuse? When the triangle is obtuse, the three altitudes don't intersect. But do you think they would if they were long enough? Follow the steps below to investigate that question. 13. Make sure the triangle is obtuse. Construct two lines that each contain an altitude. their point of intersection. ftt+ Construct This point is called the orthocenter of tnen, I ":J;[:?"X"??3: ,'"::;:*:jJ:^" oI Inemt in ir.," Con.i'r.i menu, choose lntersection I the triangle. I l I 15. Construct altitude. a line containing the third Steps 13 and'14 16. Drag the triangle and observe the lines. Q5 What do you notice about the lines containing the altitudes? Explore More 1. Hide everything in your sketch except the triangle and the orthocenter. Make and save a custom tool called "Orthocenter." You can use this tool in other investigations of triangle centers. 75 I . Chapter 3: Triangles Exploring Geometry with The Geometer's Sketchpad @2002 Key Curriculum Press Angle Bisectors in a Triangle Name(s): In this investigation, you'll discover some properties of angle bisectors in a triangle. and lnvestigate L. Construct kiangle ABC. 2. Construct the bisectors of two of the three angles: lA and ZB, J. Construct point D, the point of intersection of the two angie bisectors. 4. Construct the bisector of lC. QI What do you notice about Steps 1-3 this third angle bisector (not shown)? Drag each vertex of the triangle to conJirm that this observation holds for any triangle. ;: ect point o.qld b 5. Measure the distances from D to each of the three sides. ,.:ffi:li",::ll: '-: Veasure menu, | 6. Drag each vertex of the triangle and observe the distances. :-:cse Distance. | I :::eat for the other t*o-s'oei. I I I A2 The point of intersection of the angle bisectors in a triangle is called the incenter. Write a conjecture about the distances from the incenter of a triangle to the three sides. Explore More L. An inscribed circle is a circle inside a triangle that touches each of the three sides at one point. Construct an inscribed circle that stays inscribed no matter how you drag the triangle. (Hint: You'llneed to conskuct a perpendicular line.) 2. Make and save a custom tool for constructing the incenter of a triangle (with or without the inscribed circle). You can use this tool when you investigate properties of other triangle centers. 3. Explain why the intersection of the angle bisectors would be the center of the inscribed circle. Hint: Recall that any point on an angle bisector is equidistant from the two sides of the angle. why would the incenter be equidistant from the three sides of the triangle? Exploring Geometry with The Geometer's Sketchpad 2002 Key Curriculum Press Chapter 3: Triangles . 77 @ t2 The Euler Segment Narne{s}: In this investigation, you'll look for a relationslii: ar:rong four points of concurrency: the incenter, the circumcenter, the orthocenter, and the centroid. You'll use custom tools to construct these triangle centers, eith": those you made in previous investigations or pre-made tools. Sketch and lnvestigate 1. Open a sketch (or sketches) of yours that contains tools for the triangle centers: incenter, circumcenter, orthocenter, and centroid. Or open Triangle Centers.gsp. 2. Construct a triangle. 3. Use the Incenter tool on the triangle's vertices to construct its incenter. 4. If necessarf , give the incenter a label that identifies it, such as l for incenter. 5. ltl itl rl[ You need only the triangle and the incenter for now, so hide anything extra that your custom tool may have constructed (such as angle bisectors or the incircle). lfl [r 6. Use the Circumcenter tool on the same triangle. Hide any extras so that you have just the triangle, its incenter, and its circumcenter. If necessarf , give the circumcenter a label that identifies it. 7. Use the Orthocenter tool on the same triangle, hide any extras, and label the orthocenter. the Centroid tool on the same triangle, hide extras, and label the centroid. You should now have a triangle and the four triangle centers. 8. Use Q{ Dragyour triangle around and observe how the points behave. Three of the four points are always collinear. Which three? 9. Construct a segment that contains the three collinear points. This is called the Euler segment. 78 . Chapter 3: Triangles Exploring Geometry with The Geometer's Sketchpad @ 2002 Key Curriculum Press :riillDmllrll' l0 rur F- The Euler Segment (continued) Q2 Drag the triangle again and rook for interesting rerationships on the to check special oiu"giZr,luch as isosceles and *1";'_::"r::i right triangles. I:'_"1" D1s5r1be u.,y ,p".iur ,ri"ffiJi";h,.il fi"T,H;": centers are related in interesting wuys or located in interesting praces. *' which of the three points are arways endpoints of the Eurer segment and which point is always between them? Measure the distances arong the two parts of the Eurer segment. Drag the triangle and rook for a rerationship between these rengths. are the lengths of the two parts of the Eurer segment rerated? I"* Test your conjecture using the ialculator. Explore More 1. 2. Construct a circre centered at the midpoint of the Euler segment and passing through the midpoint of or," of the sides of the triangre. This circle is calted the nine-point circre.The,miJpoi.rt i;;;;r", through is one of the nine points. iArhat are the other eigntl Gii;;,six of them have to do with the altitudes and the orthocJnt".) once you've constructed thl nine-point circle, drag your triangre around and investigate speciar triangies. Describ" ilt tuiangres in which some of the nine points coinciie Lrcrclng Geometry with The Geometer,s St<etcffi E V-t:.2 Key Curriculum press Chapter 3: Triangles . 79 Excircles of a Triangle Namqs): ! You may have learned previously to inscribe a circle inside a triangle: Find the intersection of the angle bisectors and use that point as the center of the circle. In this demonstration, you'll define an excircle and investigate its properties. Sketch and lnvestigate 1. Open the sketch Excircles.gsP. You'll see a small triangle with a small circle inside it (the incircle) and three larger circles outside of it (excircles)' 2. Dragany vertex of the smali triangle. Qi What do You notice about the incircie? {I a2 What do You notice about each excircle? To better understand what an excircle is, you should experiment constructing circles inside angles. Follow the steps beiow. with 3. Open a new sketch, but leave the excircles sketch oPen. Press and hold down the mouse button on the Segment tool to show the Straightedge palette. Drag right to choose the RaY tool. 4. Use the RaY tool to construct an angle BAC, as shown. to construct a circle that is tangent to both sides of the angle. 5. Experiment to discover a waY Q3 Where must the center of the circle be located? rl 80 . Chapter 3: Triangles Exploring Geometry with The Geometer's Sketchpad @2002 KeY Curriculum Press Excircles of a Triangle (continued) , __o !-rr4.6o or".rr:r, mind what you may f,u.r" xeeptng in lrrt qrscovered about circles ,:1* a ray from one vertex of the ""d ""?i;;. triangle tfuough the center ot the excircles. of anr. Qa What is special about this ray? How do you know? 8. Draw-rays from!r lqLrr each vertex ve.ffex oI of the th, friangle excircle center. 9. Q5 through each v.ertices and observe retationships in the figure. as you can about relationships "langte f::,: as many Write cor'qecfures this sketch. sketch. in Explore More t ff:#riff*tilJ:r"ffite this sketch from scratch. For hints, rry showing lxptoring ceomerry *itn ii-.-.--.------.--.---.--.------.--n r's sketch pad g 2oo2 Key cu rricur um rr"l", "ot "t" Chapter 3: Triangles .11 The Surfer and the Spotter Name(s): Two shipwreck survivors manage to swim to a desert island. As it happens, the shape of the island is a perfect equilateral triangle. The survivors have very different dispositions. Sarah soon discovers that the surfing is outstanding on all three of the island's coasts, so she crafts a surfboard from a fallen tree. Because there is plenty of food on the island, she's content to stay on the island and surf for the rest of her days. Spencer, on the other hand, is more a social animal and sorely misses civilization. Every day he goes to a different corner of the island and searches the waters for passing ships. Each castaway wants to locate a home in the place that best suits his or her needs. They have no interest in living in the same place, though if it turns out to be advantageous, neither is against the idea either. Sarah wants to visit each beach with equal frequency, so she wants to find the spot that minimizes the total length of the three paths from home to the three sides of the island. Spencer wants his house to be situated so that the total length of the three paths from his home to the three corners of the island is minimized. Where should they locate their huts? llXlllillflrl0l[ 1lilll0t ,llllii rlllt Sketch and lnvestigate Use a custom tool or I 1. construct the triangle l? from scratch. ] Select point E and the three sides. Then, in the Construct menu, choose Perpendicular Construct an equilateral triangle ABC. \-- 2. Construct DA, DB, and.DC, where D is any point inside the triangle. Point D represents Spencer's hut and the segments represent paths to the corners of the island. J. Construct point E anywhere inside the triangle. 4. Construct lines through point E perpendicular to each of the three sides of the triangle. 5. Lines. You'll get all three perpendicular Construct ff, F,G, and.EE, where F, G, andH are the points where the perpendiculars intersect the sides o{ the triangle. lines. 6. Hide the perpendicular lines. Point E represents Sarah's hut and the segments from it represent the paths from her hut to the beaches. i= 71' Measure DA, DB, and DC and calculateDA + DB + DC. lmenu to ooen the Calcutatcir, Click | 8. Measure EF, EG, andEH and calculate EF + EG + EH. Choose Calculate fromtheMeasure I once on a measurement to enter tt tnto a calculation. I II o/. I Q1 82 . Chapter 3: Triangles Move points D and E (Spencer and Sarah) around inside your triangle. See if you can find the best location for each castaway. What are the best locations for Spencer's and Sarah's huts? On a separate sheet, explain why these are the best locations. Exploring Geometry with The Geometer's Sketchpad @ 2002 Key Curriculum Press :-ji: : -i- Morley's Theorem Name(s): You may know that it's impossible to trisect an angie with compass and straightedge. Sketchpad, however, makes it easy to trisect an angle. In this investigation, you'Il trisect the three angles in a triangle and discover a surprising fact about the intersections of these angle trisections. Sketch and lnvestigate L. Use the Ray tool to construct triangle ABC, drawing your rays in counterclockwise order as shown. 2. Measure Z"BAC. J. Use the Calculator to create an expression for wIBAC / 3. 4. Mark point A 5. Mark the angle measurement as a center for rotation. Step 1 mtBAC/3. 6. Rotate ray AB by the marked angle. Rotate it again to trisect ICAB. nIACB =73.1" ml *-3AC = 66.6o -_tsAC = 22.2o Steps 2-6 Step 7 7. Measure the other two angles and repeat steps 3 through 6 on those angles to trisect them. 8. Morley's theorem states that certain intersections of these angle trisectors form an equilateral triangle. Can you find it? Drag vertices of the triangle and watch the intersections of the trisectors. llr:r':.-g i Geometry with The Geometer's Sketchpad ''-t:2 <ey Curriculum Press Chapter 3: Triangles . 83 Morley's Theorem (continued) lt 9 When you think you know which intersections form an equilateral triangle, construct those intersection points and the equilateral ""0._"]::,ll:* | c""rirr"ir".r,l triangle'sinterior. choose Triangle ,",",,9"?lltj'"1H: lnterior' 10. Drag to confirm that you've constructed the triangle at the correct I intersections. If you can't tell for sure by looking, make the measurements necessary to confirm that the triangle is equilateral. Qi State Morley's theorem. Explore More 84 . if you can find other relationships or special triangles in your figure. L. See 2. Construct rays from the vertices of your original triangle through opposite vertices of the equilateral triangle. What do you notice? J. Construct a triangle using lines instead of rays. Trisect one set of exterior angles. Can you find an equilateral triangle among the intersections of these trisectors? Chapter 3: Triangles Exploring Geometry with The Geometer's @ 2OO2 Key Curriculum Napoleon's Theorem Name(s): French emperor Napoleon Bonaparte fancied himself as something of an amateur geometer and liked to hang out with mathematicians. The theorem you'll investigate in this activity is attributeci to him. Sketch and lnvestigate 1. Construct an equilateral triangle. You can use a pre-made custom tool or construct the triangle from scratch. One way to rmns-ct the center $ E :onstruct two rre: ans and their Construct the center of the triangle. ;@m :'intersection. Hide anything extra you may have constructed to construct the triangle and its center so that you're left with a figure like the one shown at right. -i;-ect the entire I 'ro,re: then choose l-' Arlte New Tool ;:rn the Custom Tools menu n the Toolbox i-e bottom tool). I I I I I 4. Make a custom Next, you'lluse your cu stom tool to construct equilateral triangles on the sides of an arbitrary tria ngle. 5. Open a new sketch. 6. 3e sure to attach :ach equilateral r-a gle to a pair of > 7. :riangle ABC's ..,ertices. lf your *:-riateral triangle goes the wrong aay (overlaps the -'.erior ol LABQ :r is not attached properly, undo ard try attaching it again. tool for this construction. Construct LABC. Use the custom tool to construct equilateral triangles on each side of A,ABC. 8. Drag to make sure each equilateral triangle is stuck to a side. 9. Construct segments connecting the centers of the equilateral triangles. 10. Drag the vertices of the original triangle and observe the triangle formed by the centers of the equilateral triangles. This triangle is called the outer Napoleon triangle of LABC. Q{ State what you think Napoleon's theorem might be. Explore More 1. Construct segments connecting each vertex of your original triangle with the most remote vertex of the equilateral triangle on the opposite side. What can you say about these three segments? -ploring Geometry with The Geometer's Sketchpad Q ffi2 Key Curriculum Press Chapter 3: Triangles . 85