points of concurrency
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Points of Concurrency. MM1G3e Students will be able to find and use points of concurrency in triangles. Median of a Triangle. A segment from one vertex of the triangle to the midpoint of the opposite side. How many medians does a triangle have?. - PowerPoint PPT PresentationTRANSCRIPT
Points of Concurrency
MM1G3e Students will be able to find and use
points of concurrency in triangles.
Median of a Triangle
• A segment from one vertex of the triangle to the midpoint of the opposite side.
The intersection of the medians is called the CENTROID.
How many medians does a triangle have?
Theorem 5.8
The length of the segment from the vertex to the
centroid is twice the length of the segment from the centroid to the midpoint.
A BF
X
E
C
D
A BF
X
E
C
D
In ABC, AN, BP, and CM are medians.
A
B
M
P E
C
NIf EM = 3, find EC.EC = 2(3)
Ex: 1
EC = 6
In ABC, AN, BP, and CM are medians.
A
B
M
P E
C
NIf EN = 12, find AN.AE = 2(12)=24
Ex: 2
AN = 36
AN = AE + ENAN = 24 + 12
In ABC, AN, BP, and CM are medians.
A
B
M
P E
C
N
If CM = 3x + 6, and CE = x + 12, what is x?CM = CE + EM
Ex: 3
x = 8
3x + 6 = (x + 12) + .5(x + 12)3x + 6 = x + 12 + .5x + 63x + 6 = 1.5x + 18
1.5x = 12
Altitude
The intersection of the altitudes is called the ORTHOCENTER.
How many altitudes does a triangle have?
Tell whether each red segment is an altitude of the triangle.
The altitude is the “true height” of
the triangle.
In ABC, CE and AD are medians.A E
C
B
DG
1. If CD = 3.25, what is BC?
2. Find AG if DG = 10.
3. If CG = 7, find CE?
Altitude, perpendicular bisector, both, or neither?
6.5
20
10.5
ALTITUDE
NEITHERBOTH
PER. BISECTOR
Homework Answerspage 280 1-6, 10-14
1. 82. 163. 54. 155. 126. 6
10. Yes, yes, yes11. No, no, no12. No, yes, no13. 12, 78o
14. 6.5, 15
The intersection of the perpendicular bisector is called the CIRCUMCENTER.
How many perpendicular
bisectors does a triangle have?
What is special about the
CIRCUMCENTER?
Equidistant to the vertices of the triangle.
Example 1:Point G is the circumcenter of the triangle. Find GB.
B
A
C
G
ED
F
2
5
7
GB=7
Example 2:Point G is the circumcenter of the triangle. Find CG.
B
A
C
G
ED
F
6
8
CG=10
Angle Bisector
The intersection of the angle bisectors is called the INCENTER.
How many angle bisectors does a triangle have?
What is special about the INCENTER?
Equidistant to sides of the triangle
Example 1:Point N is the incenter of the triangle. Find the length of segment ON.
ON=18
30 18
Example 2:Point N is the incenter of the triangle. Find the length of segment NP.
NP=15
p. 266 #13-18
p. 275 #14-17