triangles: points of concurrency mm1g3 e. investigate points of concurrency ...
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Triangles: Points of Concurrency
MM1G3 e
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Investigate Points of Concurrency
• http://www.geogebra.org/en/upload/files/english/Cullen_Stevens/trianglecenters.html
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Circumcenter
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Perpendicular Bisectors and Circumcenters Examples
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A perpendicular bisector of a triangle is a line or line segment that forms a right angle with one side of the triangle at the midpoint of that side. In other words, the line or line segment will be both perpendicular to a side as well as a bisector of the side.
DCB
A
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A
BCD
E
F
it will. sometimes however,
a vertex; throughgo obisector tlar perpendicua for necessary
not isIt vertices. theofany containnot does that Notice*
. ofbisector lar perpendicua is
EF
ABCEF
. that Notice
side.a bisects also only However,
. of side one lar toperpendicu are and Both
BFAF
EF
ABCEFAD
Example 1:
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Example 2:
Q
P N
M
€
In ΔMNP above, MQ ≅ NQ and PQ⊥MN .
Therefore, PQ is the perpendicular bisector of MN .
*Notice that PQ contains the vertex P.
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Since a triangle has three sides, it will have three perpendicular bisectors. These perpendicular bisectors will meet at a common point – the circumcenter.
FE
G
D
G is the circumcenter of ∆DEF.
Notice that the vertices of the triangle (D, E, and F) are also points on the circle. The circumcenter, G, is equidistant to the vertices.
. So, FGEGDG
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The circumcenter will be located inside an acute triangle (fig.1), outside an obtuse triangle (fig. 2), and on a right triangle (fig. 3). In the triangles below, all lines are perpendicular bisectors. The red dots indicate the circumcenters.
fig. 1 fig. 3
fig. 2
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Example 3: A company plans to build a new distribution center that is convenient to three of its major clients, as shown below. Why would placing this distribution center at the circumcenter be a good idea?
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The circumcenter is equidistant to all three vertices of a triangle. If the distribution center is built at the circumcenter, C, the time spent delivering goods to the three major clients would be the same.
C
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In Summary
• The circumcenter is the point where the three perpendicular bisectors of a triangle intersect.
• The circumcenter can be inside, outside, or on the triangle.
• The circumcenter is equidistant from the vertices of the triangle
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Circumcenter
• Exploration
• Construction
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Medians and Centroids Examples
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A median of a triangle is a line segment that contains the vertex of the triangle and the midpoint of the opposite side. Therefore, the median bisects the side.
triangle. theof mediana is .
and ofmidpoint theis Therefore, . bisects above, In
ADCDBD
BCDBCADABC
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Since a triangle has three sides, it will have three medians. These medians will meet at a common point – the centroid.
. of centroid theis
.point at intersect medians
The triangle. theof medians are
and , , that see can we
, on markings theFrom
ABCO
O
CFBEAD
ABC
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The centroid is always located inside the triangle.
Acute triangle
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The distance from any vertex to the centroid is 2/3 the length of the median. Q
RSD
E FG
8
1232
32
.12 Suppose . of length the
of 32 is centroid the to vertex from
distance The median.a is , In
QG
QG
QDQG
QDQD
GQ
QDQRS
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Example 1: G is the centroid of triangle QRS. QG = 10 GF = 3. Find QD and SF.
Q
RSD
E FG
QD
QD
QDQG
153
210
32
SF
SF
SFGFSFSG
9
3
. then, Since
31
31
32
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Example 3: G is the centroid of triangle DEF. FG = 15, ES = 21, QG = 5 Determine FR, EG and GD
Q
R
S
D
E
F G
FR
FR
FRFG
5.22
15 32
32
15
21
5
14
2132
32
EG
EG
ESEG
10
52
.2 then
,
and Since
31
32
GD
GD
QGGD
QDQG
QDGD
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Notice that the distance from any vertex to the centroid is 2/3 the length of the median. That means that the distance from the centroid to the midpoint of the opposite side is 1/3 the length of the median.
So, in triangle MNP, MQ=2(QT) and QT=(1/2)MQ
Q
P
N
U
M
T
V
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The centroid is also known as the balancing point (center of gravity) of a triangle.
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In Summary• A median is a line segment from the a vertex of a triangle
to the midpoint of the opposite side.
• The distance from the vertex to the centroid is 2/3 the length of the median.
• The distance from the centroid to the midpoint is 1/3 the length of the median, or half the distance from the vertex to the centroid.
• Since the centroid is the balancing point of the triangle, any triangular item that is hung by its centroid will balance.
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Centroids
• Investigate
• Construction
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Angle Bisectors and Incenters Examples
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An angle bisector of a triangle is a segment that shares a common endpoint with an angle and divides the angle into two equal parts.
triangle. theofbisector angle an is Therefore
. so parts equal twointo divides , In
AD
CADBADBACADABC
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Example 1: Determine any angle bisectors of triangle ABC.
A
BC
D
E
FG
triangle. theof
bisector angle an is Therefore,
. that see can we
triangle, theon markings theFrom
BG
CBGABG
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Since a triangle has three angles, it will have three angle bisectors. These angle bisectors will meet at a common point – the incenter.
M
ZY
X
triangle. theofincenter theis Therefore,
.point at intersect bisectors angle three the, In
M
MXYZ
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The incenter is always located inside the triangle.
Acute triangle
Right triangle
Obtuse triangle
incenter
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The incenter is equidistant to the sides of the triangle.
triangle. theof sides theof each
from , distance, same theis
incenter. theis , on,intersecti
ofpoint Thebisectors. angle are
and ,,, In
xG
G
CFBEADABC
x
x
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Example 2: L is the incenter of triangle ABC. Which segments are congruent?
L
A
B
C
D
EF
. Therefore,
triangle. theof sides the topoint
from distance therepresent and
,, segments,lar perpendicu
The triangle. theof sides the
t toequidistan isincenter The
LFLELD
L
LF
LELD
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Example 3: Given P is the incenter of triangle RST. PN = 10 and MT = 12, find PM and PT.
Not drawn to scale
10
12
Theorem. an Pythagoreby the
find can wengle,right triaa is
since Also, .10
So, .triangle,
theofincenter theis Since
PT
PMTPM
PMPN
P
PT
PT
PT
PTMTPM
612
244
1210
.
2
222
222
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In Summary
• The incenter is the point of intersection of the three angle bisectors of a triangle.
• The incenter is equidistant to all three sides of the triangle.
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Incenter
• Investigate
• Construction
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Altitudes and Orthocenters Examples
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An altitude of a triangle is the perpendicular distance between a vertex and the opposite side. This distance is also known as the height of the triangle.
D C
B
A
. vertex thecontains
and lar toperpendicu is that Notice . of altitude an is
A
BCADABCAD
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Example 1: Determine any altitudes of triangle ABC.
FE
D
C
BA
. side, opposite thelar toperpendicunot isit since altitude annot is
. vertex, thecontains and lar toperpendicu is . vertex, thecontains
and lar toperpendicu is altitudes. are and only , In
ACBE
CABCDA
BCAFCDAFABC
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Since a triangle has three sides, it will have three altitudes. These altitudes will meet at a common point – the orthocenter.
O
Z
Y
X
. ofr orthocente theis
. point, common at themeet altitudes These
triangle. theof altitude an each are and ,, that Notice
ABCO
O
CXBYAZ
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The orthocenter may be located inside an acute triangle (fig. 1), outside an obtuse triangle (fig. 2), or on a right triangle (fig. 3). In the triangles below, the red lines represent altitudes. The red points indicate the orthocenters.
Obtuse TriangleAcute Triangle Right Triangle
Fig. 3Fig. 2Fig. 1
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Summary• An altitude is a line segment containing a
vertex of a triangle and is perpendicular to the opposite side.
• The orthocenter is the intersection point of the three altitudes of a triangle.
• Orthocenters can be inside, outside, or on the triangle depending on the type of triangle.
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Orthocenter
• Investigate
• Construction
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Points of Concurrency
Investigate
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Points of Concurrency
MM1G3 eReview
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B
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D
![Page 65: Triangles: Points of Concurrency MM1G3 e. Investigate Points of Concurrency glish/Cullen_Stevens/trianglecenters.html](https://reader036.vdocuments.net/reader036/viewer/2022062423/5697bfdb1a28abf838cb07ca/html5/thumbnails/65.jpg)
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B
![Page 66: Triangles: Points of Concurrency MM1G3 e. Investigate Points of Concurrency glish/Cullen_Stevens/trianglecenters.html](https://reader036.vdocuments.net/reader036/viewer/2022062423/5697bfdb1a28abf838cb07ca/html5/thumbnails/66.jpg)
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A
![Page 67: Triangles: Points of Concurrency MM1G3 e. Investigate Points of Concurrency glish/Cullen_Stevens/trianglecenters.html](https://reader036.vdocuments.net/reader036/viewer/2022062423/5697bfdb1a28abf838cb07ca/html5/thumbnails/67.jpg)
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A
![Page 68: Triangles: Points of Concurrency MM1G3 e. Investigate Points of Concurrency glish/Cullen_Stevens/trianglecenters.html](https://reader036.vdocuments.net/reader036/viewer/2022062423/5697bfdb1a28abf838cb07ca/html5/thumbnails/68.jpg)
Try These: B
![Page 69: Triangles: Points of Concurrency MM1G3 e. Investigate Points of Concurrency glish/Cullen_Stevens/trianglecenters.html](https://reader036.vdocuments.net/reader036/viewer/2022062423/5697bfdb1a28abf838cb07ca/html5/thumbnails/69.jpg)
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D
![Page 70: Triangles: Points of Concurrency MM1G3 e. Investigate Points of Concurrency glish/Cullen_Stevens/trianglecenters.html](https://reader036.vdocuments.net/reader036/viewer/2022062423/5697bfdb1a28abf838cb07ca/html5/thumbnails/70.jpg)
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A
![Page 71: Triangles: Points of Concurrency MM1G3 e. Investigate Points of Concurrency glish/Cullen_Stevens/trianglecenters.html](https://reader036.vdocuments.net/reader036/viewer/2022062423/5697bfdb1a28abf838cb07ca/html5/thumbnails/71.jpg)
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C