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. BD 2. Drag different vertices of h--.- \ each of the four triangles / \ t\ to observe and compare AL___\c yle how the triangles behave. QI Which of the kiangles LIqA seems the most flexible? ------__. \ ,/ \ Explain. * M\----"--: "<-----\U \

a2 \Alhich of the triangles seems the least flexible? Explain.

-: measure each I - :-f le, select three l? 3. Measure the three angles in AABC. coints, with the rei€x I loUr middle I 4. Draga vertex of LABC and observe the changing angle measures. To Then, in =ection. I le r/easure menu, answer the following questions, note how each angle can change from I :hoose Angle. I being acute to being right or obtuse.

Q3 How many acute angles can a triangle have? ;;,i:?:PJi;il: I qo How many obtuse angles can a triangle have? to classify -sed I rc:s. These terms f> Qg LABC can be an obtuse triangle or an acute :"a- arso be used to I

: assify triangles I triangle. One other triangle in the sketch can according to I also be either acute or obtuse. Which triangle is it? their angles. I aG Which triangle is always a right triangle, no matter what you drag? 07 Which triangle is always an equiangular triangle? -: -Fisure a tensth, I =ect a seoment. I thenlin t'e p t,'leasure.. 5. Measure the lengths of the three sides of LABC. menu, I :-coselength.l z n t t a An- 1 r i, r . 6. Drag a vertex of LABC and observe the changing side lengths.

furonng Geometry with The Geometer's Sketchpad Chapter 3: Triangles . 63 tory2 Key Curriculum Press { Defining Triangles (continued)

Scalene. isosceles, l-9 none the side lengths are equal, the triangle and equilateral are aB If of I terms used to I is a scnlene triangle. If two or more of the sides classify triangles by I are equal in length, the triangle is isosceles.lf all relationships among I their sides. Because I three sides are equal in length, itis equilaternl. AABC has no I constraints, it can be I Which type of triangle is AABC most of the time?

any type of triangle. 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 EXplOre MOre to get an idea of I how the triangles in the Glassify 1. Open a new sketch and see if you can come up with ways to construct Triangles.gsp a right triangle, an isosceles triangle, and an equilateral triangle. sketch were constructed. Describe your methods.

64 . Chapter 3: Triangles Exploring Geometry with The Geometer's Sketchpad @ 2002 Key Curriculum Press friangte Sum Name(s): conjectureabout":::'**'i?:i':::.1*'rrinvestiga*FT:^1-: two-part invesrigation. First you,rr";;t;;,k"ff - To measure l"T;jliffi an J :ill:,#*trq*,fl sketch and t*T,J:",n"" ;1?',3;:frilfrii rnvestisate *,fi3j*,J5ifu/ 1 Construct AAB]. choose anere l-t 2' M"usure its ChooseCalculate three q! (6rsD. I 'uuse sdlcUlate - -- angres. itx:E::i::t tz-/tEi(j / = 87.2 Calculate mZBcA= 4e'5o t{ff,$idffi f : the sum of the angle measures. once on u I +. Draga vertex *r' / *of tlthe triangte and n*== '.".,,".1nitJunr"iii''io i il:;;::::the angle sum. observe calcutation, 1 J _ seiecr ai What is the sum ooint B I orF the angles in any Follow fuianele? ,"?#43#:i/ these steps to investigate why,J'ff;:l";; s. Consrructr.-orsrruct a tine through point " .;;";:":ilf;:;:: I : B paraltelparalret to At. " 6' Construct the midpoints cnoose of A-B and CE. "f"j,"t.lii*il,:Triangls l=i _ Lii.i.',-'.i 7. Consfruct F the interio r of AABC.

enter ror ,?t?,*jffi ro ta ti on and rota lransform ?ii,ffi te the men,, I[',Hi'?81'r,f""#ii ;c choose Rotate' 9' 1 Give the new fuiangle interior a different B:Fil'#'lf ( color. 'o' as a center and rotate Y:* ffi"J:,T,:idpoint the interior by 180" 11. Give this new tuiangle interior a different color. B

72. the three triangres 3,ff ilffit f,:f,|l',iffiff* are rerated to *:t:,lixftI;j*:ll#;;r?f each u--' '"vtd'ut now j#:",*d,..ne.r,he conjecfure from e1. i:f ,Tl,this 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. nICAB = 45.5o Hold down the nZABC = 67.2" mouse button on the 2. Construct AC--) to extend side AC. Segment tool and drag right to choose mtBCD=112.74 the Ray tool. 3. Construct point D onA-, outside of the triangle.

Select, in order, A C D' points B, C, and D. Measure exterior angle BCD. Then, in the Measure menu, Measure the remote interior angles ZABC and ICAB. choose Angle. 6. Drag parts of the triangle and look for a relationship between the measures of the remote interior angles and the exterior angle. Choose Galculate I ["#']H$%Tiil' f'qr How are the measures of the remote interior angles related to the Calculator. Click I measure of the exterior angle? Use the Calculator to create an once on a I measurement to I expression that confirms your conjechrre. enter it into a I

calculation. I

I

rt

I Select the interior; ( marked vector. il then, in the I Transform menu, | 10. Give the new triangle interior choose Translate. I a different color.

11,. Construct the midpoint of BC

Double-clickthelrrn as a and rotate the triangle point to mark it as a( tz' Mark the midpoint center for rotation center. Select the I interior about this point by 180". interior; then, in the I Transfrom menu, I n. choose Rotate. IJ. Give this new triangle interior a different color. 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 Name(s): In this investigation, you'll discover relationships among the measures of t the sides and angles in a triangle.

Sketch and lnvestigate

1. Construct a triangle. -: 'reasure a side, m AB- = 2.7 cm r=.ct it; then, in the mBe=3.6cm lr4easure menu, 2. Measure the lengths of . rloose Length. the three sides. mffi=3.4cm

- -:ose Calculate '':rn the Measure 3. Calculate the sum of any m AB-+m BT = 6.31 cm -enu to open the two side lengths. lalculator. Click once on a 'neasurement to 4. Drag a vertex of the enter it into a calculation. 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 [, 5. Measure IABC, ICBA, and IACB. =-ole, select three points, with the .:1ex your middle 6. Drag the vertices of your triangle and look for relationships between =:lection. Then, in --: Measure menu, side lengths and angle measures. I choose Angle. 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.

AB longesl Largest AC longesl Largest BC longesl Largest side angle? side angle? side angle? AB shortest Smallest AC lo.tgest Smallest BZlongest Smallest side angle? side angle? side angle?

Q5 Summarize your findings from the chart as a conjecture. !

:rnloring Geometry with The Geometer's Sketchpad Chapter 3: Triangles . 67 3 20O2 Key Curriculum Press 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 see a figure like the one shown below, along with some text.

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 ,-). Go to the SAS page and make a triangle DEF with those given parts. the sketch window to Try to make a triangle that is not congruent to LABC. switch pages. 4. Experiment on each of the other pages.

Q3 Of SSS, SAS, SSA, ASA, AAS, and AAA, which combinations of corresponding parts guarantee congruence in a pair of triangles? Which do not?

68 . Chapter 3: Triangles 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 BZ c

5. Drag each vertex of your triangle to see how it behaves.

Q{ Explain why the triangle is always isosceles.

6. Hide circle AB. To measure an I - 7. Measure the three angles in the triangle. "'rgle, select three l? points, with the I .er1ex your middle I 8. Drag the vertices of your triangle and observe the angle measures. selection. Then, in I :e Measure menu, I choose a2 Angle s ACB and. ABCare the base anglesof the isosceles triangle. Angle. I 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 Chapter 3: Triangles . 69 g 2002 Key Curriculum Press 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 Name(s): A median in a triangle connects a vertex with the midpoint of the opposite t 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.

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.

' _,3u've already ::-s:ructed three 4. Construct the point of intersection of *edians, select the two medians. :.. - :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.

-sino the Text I - / ,: clrEk nn.. nn l-- o' The point where the medians :-: ooint to show I intersect is called the centroid. :s abel. Double- |

: ck the label to I Show its label and change it

change it, I to Ce for centroid.

-o measure the r stance between 7. Measure the distance from B to :,',c points, select Ce and the distance from Ce to :-em; then, in the :'/easure menu, the midpoint F. --:ose Distance. BCe CeF 8. Drag vertices of AABC and 2j7 cm 1.08 cm look for a relationship 1.00 cm 0.50 cm CeF. between BCe and 3.24 cm 1.62 cm

Select the two ?..tJlj t:ry1 1"tt7 *ns measurements; 9. Make a table with these two then, in the measures. Graph menu, 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 Chapter 3: Triangles . 71 O 2002 Key Curriculum Press f,ledians in a Triangle (continued) to l-tfZ. g*ed on what you notice about the table erttries, use the Calculator "1;;",fil".:f,J,: even ryl;ii"*liii I make an expression with the measures that n'ill remain constant Calculator."- --;;;";;; Click I I as the measures change. measurement to I ":5:[$3f I o, Write the expression you calculated in step 1'2'

median in e3 Write a conjecture about the way the centroid divides each a triangle.

the Plot 13. Plot the table data on a piece of graph Paper or using points command in thebraph menu. You should get a graph with several collinear Points.

1.4. Draworconstructalinethroughanytwoofthedatapointsand measure its sloPe. QaExplainthesignificanceoftheslopeofthelinethrough the data Points.

Explore More of a triangle' Save 1. Make a custom tool that constructs the centroid application your sketch in the Tool Folder (next to the sketchpad for future itself on your hard drive) so that the tool will be available investigations of triangle centers'

Exploring Geometry with The Geometer's . 3: Triangles 72 Chapler @2OO2 KeY Curriculum Perpendicular Bisectors in a Triangle Name(s): In this investigatiory you'rl discover properties of perpendicular bisectors in a triangle. you'rl aiso rearn how to construct a circre that passes through each vertex of a triangle. Sketch and lnvestigate

1. Construct triangle ABC.

2. Construct the midpoints of l: :ct a side and the sides. . ridpoint; then, - :he Construct 3. Construct .nenu, choose two of the three Perpendicular perpendicular bisectors Line. in the triangle. Click afthe L -'=--:ction with the I - 4. Construct point the point Selection G of Arrow I :col or intersection of these lines. with the I

Point toot. i 5. Construct the third perpendicular bisector. Q{ what do you notice about this third perpendicurar bisector (not Drag a vertex :h?y")? of the trianglea to conJirm that this conjecture holds for any triangle.

6. 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 triangle? the

Q3 Drag a vertex until point G falls on a side of the triangle. What triangle kind of is this? Where exactly does point G lie?

:= ect point G and a 7. .ertex; then, in the Measure the distances from point G to each Measure menu, of the three vertices. :.oose Distance. ::peat 8. Drag a vertex for the other of the triangre and observe the distances. two vertices. 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? 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 a custom 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. 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?

J. See if you can circumscribe other shapes besides triangles. Describe what you try and include below any additional conjectures you come up with.

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 B.

Ql 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 4 Steps 5 and 6

7. Drag vertices of the triangle and observe how your altitude behaves.

Q2 Where is your altitude when lAis a right angle?

AiltilMrr,l Geometry with The Geometer's Sketchpad Chapter 3: Triangles . 75 F tr1li;:

Select evervthino in vori .r"i.'n. iii"n, | 8. Drag your kiangle so that it is acute again (with the altitude falling press the Gustom I tools icon (the inside the triangle). I

bottom tool in the I Toolbox) and choose f> p. Make a custom tool for this construction. Name the tool "Altitude." Greate New Tool l

from the menu I il;i;;#;i; I 10 Use your custom tool on the triangle's vertices to construct a second Starting with three altitude. Don't worry if you accidentally construct the altitude that points (the tool's already exists. use the tool on the vertices again in a "givens"), the Just different Altitude tool will order until you get another altitude. construct a triangle and an altitude from one of the vertices. 11. Use your Altitude tool to construct the To use the tool, click third altitude in the triangle. on the Custom tools icon to select the most recently 12. Drag the triangle and observe how the created tool, then click on the three three altitudes behave. triangle vertices. 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. Construct their point of intersection. ":J;[:?"X"??3: ftt+ ,'"::;:*:jJ:^" This point is called the orthocenter of oI Inemt tnen, I I in ir.," Con.i'r.i I the triangle. menu, choose lntersection I l 15. Construct a line containing the third altitude. 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 . Chapter 3: Triangles Exploring Geometry with The Geometer's Sketchpad @2002 Key Curriculum Press I 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 this third angle bisector (not Steps 1-3 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: I Veasure menu, | 6. Drag each vertex of the triangle and observe the distances. :-:cse Distance. :::eat I for the other I t*o-s'oei. 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 Chapter 3: Triangles . 77 @ 2002 Key Curriculum Press 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

ltl incenter. itl rl[ 5. You need only the triangle and the incenter for now, so hide anything [rlfl extra that your custom tool may have constructed (such as angle bisectors or the incircle). 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.

8. Use the Centroid tool on the same triangle, hide extras, and label the centroid. You should now have a triangle and the four triangle centers.

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 :riillDmllrll' @ 2002 Key Curriculum Press 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 right*1";'_::"r::i I:'_"1" as isosceles *'and triangles. 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 and Eurer segment 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 I"* two parts of the Eurer segment rerated? Test your conjecture using the ialculator.

Explore More

1. 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".) 2. 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

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?

a2 What do You notice about each excircle? {I

with To better understand what an excircle is, you should experiment constructing circles inside angles. Follow the steps beiow. 3. Open a new sketch, but leave the excircles sketch oPen. Press and hold down 4. Use the RaY tool to construct an the mouse button on the Segment tool angle BAC, as shown. to show the Straightedge a waY to palette. Drag right to 5. Experiment to discover choose the RaY tool. construct a circle that is tangent to both sides of the angle. Q3 Where must the center of the circle be located?

rl

Exploring Geometry with The Geometer's Sketchpad . Chapter 3: Triangles 80 @2002 KeY Curriculum Press Excircles of a Triangle (continued)

mind __o !-rr4.6o or".rr:r, what you may f,u.r" qrscovered xeeptng in lrrt about circles , ,:1* a ray from one ""d ""?i;;. vertex of the triangle tfuough ot the excircles. the center of anr. Qa What is special about this ray? How do you know?

8. Draw-rays from !r lqLrreach vertexve.ffex ofoI theth, friangle excircle center. through each

9. v.ertices and observe "langte retationships in the figure. Q5 Writef::,: as many cor'qecfures as you can thisthis sketch. about relationships in

Explore More t this sketch from ff:#riff*tilJ:r"ffite scratch. For hints, rry showing

lxptoring ceomerry *itn ii-.-.--.------.--.---.--.------.--- g n r's sketch pad 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 llXlllillflrl0l[ searches the waters for passing ships. Each castaway wants to locate a 1lilll0t ,llllii 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 rlllt 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?

Sketch and lnvestigate

Use a custom tool or I - 1. an triangle ABC. construct the triangle l? Construct equilateral

from scratch. ] 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.

Select point E E and the three 4. Construct lines through point sides. Then, in perpendicular to each of the three sides of the triangle. the Construct menu, choose Perpendicular 5. Construct ff, F,G, and.EE, where F, G, andH are the points where the Lines. You'll get all three perpendicular 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. Choose Calculate i= 7 + + fromtheMeasure l- 1' Measure DA, DB, and DC and calculateDA DB DC.

menu to ooen the I Calcutatcir, Click | 8. Measure EF, EG, andEH and calculate EF + EG + EH. once on a I measurement to I o enter tt tnto a I /. Move points D and E (Spencer and Sarah) around inside your triangle. calculation. I See if you can find the best location for each castaway.

Q1 What are the best locations for Spencer's and Sarah's huts? On a separate sheet, explain why these are the best locations.

82 . Chapter 3: Triangles Exploring Geometry with The Geometer's Sketchpad :-ji: : -i- @ 2002 Key Curriculum Press 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 counter- clockwise order as shown.

2. Measure Z"BAC.

J. Use the Calculator to create an expression for wIBAC / 3. 4. Mark point A as a center for rotation.

Step 1 5. Mark the angle measurement 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 Geometry with The Geometer's Sketchpad Chapter 3: Triangles . 83 i ''-t:2

,",",,9"?lltj'"1H: lt 9 When you think you know which intersections form an equilateral ""0._"]::,ll:* | triangle, construct those intersection points and the equilateral c""rirr"ir".r,l triangle'sinterior. choose Triangle I lnterior' 10. Drag to confirm that you've constructed the triangle at the correct 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.

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L. See if you can find other relationships or special triangles in your figure.

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?

84 . 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 One way to or construct the triangle from scratch. rmns-ct the center $ E :onstruct two Construct the center of the triangle. rre: ans and their ;@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 - 4. 'ro,re: then choose l-' Make a custom tool for this construction. Arlte New Tool I ;:rn the Custom I Next, you'lluse your custom tool to construct equilateral triangles on the Tools menu I sides of an n the Toolbox I arbitrary triangle. i-e bottom tool). I 5. Open a new sketch.

6. Construct LABC.

3e sure to attach :ach equilateral > 7. Use the custom tool to construct r-a gle to a pair of :riangle ABC's equilateral triangles on each side ..,ertices. lf your of A,ABC. *:-riateral triangle goes the wrong aay (overlaps the 8. Drag to make sure each equilateral -'.erior ol LABQ triangle is stuck to a side. :r is not attached properly, undo ard try attaching 9. Construct segments connecting the it again. 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.

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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 Chapter 3: Triangles . 85 Q ffi2 Key Curriculum Press