Warm-Up

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Warm-Up

• Take out homework p.301 (4-20) even
• Check your answers with a partner

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Medians
Picture
Both sides are congruent
Median
vertex to midpoint
9
How many medians can a triangle have?
Median
vertex to midpoint
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Midpoint-

• If you have a midpoint- then the segments on both
  sides are CONGRUENT!
• That is why you will see the tick marks
• For the measure of the entire line- add both
  sides!

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M
D
P
C
9
N
1. What is NC if NP 18?
2. If DP 7.5, find MP.
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12
AD and EB are medians
A
B
C
D
14
E
1.What is ED if DC 14? 2.What Is AC is BC is 9?
3.If BC 3, find AC.
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13

• So if you are given the length of the entire
  side, how do you find a missing segment?
• If you are given the length of a segment, how to
  you find the entire side?
• If you have a median- what do you know about each
  side?

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A
E
B
D
C
If CD 2x 5, BD 4x 1, and AE 5x 2, find
BE.
BD CD
AE BE
BE 13
4x 1 2x 5
BE 5x 2
BE 5(3) 2
2x 6
x 3
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The intersection of the medians is called the
CENTROID.
Draw the Picture
How many medians does a triangle have?
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Concurrent

• When three or more lines or segments meet at the
  same point.

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Theorem 6-1 Distance from vertex to centroid is
twice the distance from centroid to midpoint.
Draw Picture
2x
x
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Vertex to Centroid ? LONGER
Centroid to Midpoint ? shorter
x 2x whole median
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How much is CX?
C
D
CX 2(XF)
E
X
CX 2(13)
13
B
A
F
CX 26
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Quick Assessment

• What is a median?
• What is a centroid?
• What does concurrent mean?
• What is the vertex?
• What is a midpoint?

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How much is XD?
C
D
AX 2(XD)
E
X
18
18 2(XD)
B
A
F
9 XD
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In ?ABC, AN, BP, and CM are medians.
Ex 1
If EM 3, find EC.
C
N
EC 2(3)
P
E
EC 6
B
M
A
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In ?ABC, AN, BP, and CM are medians.
Ex 2
C
If EN 12, find AN.
N
AE 2(12)24
P
E
AN AE EN
B
AN 24 12
M
A
AN 36
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Altitude
Picture
Altitude
vertex to opposite side and perpendicular
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Altitude
The altitude is the true height of the triangle.
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Tell whether each red segment is an altitude of
the triangle.
The altitude is the true height of the triangle.
YES
NO
YES
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Perpendicular Bisector
Both sides are congruent- make sure you see this
or it is NOT a perpendicular bisector
Picture
Perpendicular Bisector
midpoint and perpendicular
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Tell whether each red segment is an perpendicular
bisector of the triangle.
NO
NO
YES
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Can you have both?

• Can it be both an altitude and perpendicular
  bisector?

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Quick Assessment

• What is the difference between altitude and
  perpendicular bisector?
• What is an altitude?
• What is a perpendicular bisector?
• How can the segment be both- altitude and
  perpendicular bisector?

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• Angle bisector of a triangle is the bisector of
  each angle of the triangle. (cuts each angle in
  half)
• Incenter of the triangle where the angle
  bisectors meet.

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Angle Bisectors
Incenter
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Concurrency of AngleBisectors of a Triangle
The angle bisectors of a triangle intersect at a
point that is equidistant from the sides of the
triangle.
B
D
PD PE PF
F
P
C
E
A