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Congruent Triangles
4.52 Importance of concurrency
TrianglePoints of Concurrency
#1Perpendicular bisectors
and the circumcenter.
C
T
A
B
3PerpBisectors
Before we can talk about the circumcenter’s importance, we need some review on perpendicular bisectors.
Circumcenter
BA
M BA
Review of Perpendicular Bisector Properties.
M BA
Every point on the perpendicular bisector is equidistant from A and B.
C
T
A
B
3PerpBisectors Therefore, the circumcenter is equidistant from
each vertex.
Why ?
C
T
A
B
3PerpBisectors
Since C is on each perpendicular bisector it is equidistant from each segment’s endpoint.
Therefore…
C
T
A
B
3PerpBisectorsThe equal distances are radii for a circle that is …
written around (circumscribed) the circle.
Ergo the term… Circumcenter.
C
T
A
B
3PerpBisectorsWhy is this important?
If A, B, and T are three cities, Point C is the ideal place to build a communication tower to broadcast to each city.
TrianglePoints of Concurrency
#2Angle bisectors
and the Incenter.
Angle Bisector
A
B C
Angle Bisector
A
B C
From any general point on the angle bisector, the perpendicular distance to either side is the same.
Why ?P
XY
Angle Bisector
A
B C
The top angles are congruent by definition of angle bisector.
There are two right angles are congruent by def. of perpendicular.
P
AP AP by the reflexive property of =
?? PAX PAY By AAS
XY
PX PY By CPCTC
Angle Bisector
A
B C
Each new point creates 2 congruent triangles by AAS.
Why is this important ?
Let’s see.
AngleBisectors
A
B C
P
Since the incenter P is on each angle bisector….
AngleBisectors
A
B C
P
Point P is equidistant from each side.
Remember, the distance from
a point to a line is the brown perpendicular segment.
Why is this important?
Because these equal distances
are radial distances.
AngleBisectors
A
B C
P
The incenter generates an inscribed circle.
Radial distances refer to a circle.
TrianglePoints of Concurrency
#3Medians
and the Centroid.
A
BC
A median is a segment connecting the vertex to the midpoint of a side of a triangle.
A
BC
P
The 3 medians meet at the centroid – point P.
EF
G
D
3 Medians Centroid
A
B C
Each little triangle is unique, yet they all have something in common.
What is it ?
Area AEF = 7.08 cm2
Area ECF = 7.08 cm2
Area EGC = 7.08 cm2
Area EBG = 7.08 cm2
Area DBE = 7.08 cm2
Area AED = 7.08 cm2
EF
G
D
3 Medians Centroid
E
F
G
D
A
B C
Since the areas of the little triangles around the centroid E are the same, …
the triangle will balance on the centroid.
Mobiles balance objects in an artistic form.
Calder’s Mobile at the East Wing of the National Gallery of ArtIn Washington DC.
Triangles balanced at their centroids.
The birds are tied to their centers of balance or centroids.
TrianglePoints of Concurrency
#4Altitudes
and the Orthocenter.
B
A
C
Acute Triangle
The point of concurrency is called…
The Orthocenter
B
A
C
Right Triangle
The point of concurrency is called…
The Orthocenter
B
A
C
Obtuse Triangle
Notice that although the altitudes are not concurrent…
B
A
C
Obtuse Triangle
The lines containing the altitudes are concurrent.
The Orthocenter
So what do you think is the importance or practical application of the orthocenter is?
Nothing !!! It is just an interesting fact that mathematicians have discovered.
Isn’t this FUN !!! Psych
Summary
bisectors
Line Type ConcurrencyPoint
Importance
bisectors
Medians
Altitudes
Circumcenter
Incenter
Centroid
Orthocenter
Equidistant from the vertices
Circumscribed Circle
Equidistant from the sides
Inscribed Circle
Center of Balance
None
C’est fini.Good day and good luck.
A Senior Citizen Production
That’s all folks.