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Warm-Up
The perpendicular bisectors meet atG. If BD = 4 and GD = 3, what is the length of GC?
Properties of Triangles – Day 3Medians and Altitudes
Theorem:Concurrency of Medians
The centroid is 2/3 the distance from each vertex to the midpoint of the opposite side.
Example 1:
D is the centroid of the triangle and BE is perpendicular to AC.
Example 2:
Draw a triangle with vertices:
D(3,6), F(7,10), and E(5,2)
Find the midpoint of each side
Find the centroid P
Theorem: Concurrency of Altitudes
The lines containing the altitudes intersect at a point called the orthocenter.
Name the line segment described: Q:Perpendicular segment from vertex to opposite side. A: Altitude Q: Segment that divides an angle of a triangle into two
congruent, adjacent angles. A: Angle Bisector Q: Perpendicular segment that intersects the side of a
triangle at its midpoint. A: Perpendicular Bisector Q: Segment connecting a vertex of a triangle to the
midpoint of the opposite side. A: Median Q: Segment that connects two midpoints of a triangle. A: Midsegment
Name the concurrent points for the following segments:
Q: Angle Bisectors A: Incenter Q: Medians A: Centroid Q: Perpendicular Bisectors A: Circumcenter Q: Altitudes A: Orthocenter
Homework – Day 3
Days 1 – 3 Review
Use your clickers to answer the following questions…
This segment’s endpoints are a vertex of a triangle and the midpoint of the opposite side.
a. Medianb. Perpendicular Bisectorc. Midsegmentd. Altitude
In WXY, Q is the centroid and YQ = 2x 15 and QA = 4. Find x.
a. 9.5b. 11.5c. 13.5
Q
Y
W
X
A
B
C
The circumcenter is equidistant to the _________ of a triangle.
a. Verticesb. Sides
In JKL, PS = 7. Find JP.
a. 7b. 14c. 21 J
K
L
R
S
T
P
This segment is perpendicular to a segment at its midpoint.
a. Medianb. Perpendicular Bisectorc. Midsegmentd. Altitude
This line passes through a vertex and divides that interior angle in half.
a. Perpendicular Bisectorb. Angle Bisectorc. Midsegment
Find the measure of KF if K is the incenter of ABC.
a. 5b. 12c. 13
A
B
C
F
D
E13
12
K
This is the intersection of the three perpendicular bisectors of a triangle and is equidistant from the vertices.a. Incenterb. Circumcenterc. Centroidd. Orthocenter
a. 2b. 4c. 6
D is the centroid of triangle ABC. Find CF.
This is the intersection of the three medians of a triangle and is 2/3 the distance from each vertex to the midpoint of the opposite side.a. Incenterb. Circumcenterc. Centroidd. Orthocenter
This is the intersection of the three angle bisectors of a triangle and is equidistant from the sides.a. Incenterb. Circumcenterc. Centroidd. Orthocenter
The incenter is equidistant to the _________ of a triangle.
a. Verticesb. Sides
This is the intersection of the three altitudes of a triangle.
a. Incenterb. Circumcenterc. Centroidd. Orthocenter
Find each measure of DC if D is the circumcenter of ABC, AD = 12, and DF = 5.
a. 5b. 12c. 13
A
B
C
DE
F
G
This is a perpendicular segment from a vertex to the opposite side.
a. Medianb. Perpendicular Bisectorc. Midsegmentd. Altitude