Download Chap 3-Triangles

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--.\
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t\
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yle
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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.
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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.
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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.
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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
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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?
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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.
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f'qr
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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.
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rt
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Select the interior;
then, in the
Transform menu,
choose Translate.
marked vector.
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il
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| 10. Give the new triangle interior
a different color.
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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,
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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.
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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,
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7. Measure the three angles in the triangle.
8. Drag the vertices of your triangle and observe the angle measures.
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choose
Angle.
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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
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-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
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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.
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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."
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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
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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