CC Geometry H
Aim #11: How do we construct a centroid and an orthocenter?
Do Now: Plot ΔABC with A(2,4), B(6,8), and C(10,0). Plot D, E, and F, the
midpoints of sides AB, BC and CA, respectively. Draw medians AE, BF, and CD.
Label centroid G, the point where the three medians intersect.
Complete:
• A median of a triangle is a segment from a
___________ of a triangle to the ____________ of the
opposite side.
• The three medians of a triangle are ______________
at a point, called the centroid. The centroid will
always be located inside the triangle.
B
D
Centroid Theorem:
The distance from a vertex to the centroid is
two-thirds the length of the median.
E
G
A
The three medians' point of concurrency is called
the centroid.
F
C
Each median of a triangle is divided into two parts by the centroid. According
to the Centroid Theorem, the ratio of the longer part to the whole median is
___:___. Therefore, the ratio of the smaller part to the whole median is
___: ___, and the ratio of the two parts (expressed from the vertex to the
centroid first) is ___: ___.
Example:
1. S, T, and R are midpoints of the sides of ΔABC.
B
a) Find TC and BC when BT = 100.
S
b) Find PS and PC when SC = 210
c) Find PA and TA when PT = 80
d) Find PR and BR when BP = 150
e) If CP = 8x - 10 and PS = 3x + 20, find CS.
f) If PT = 3y and AT = 12y - 105, find AT.
P
A
R
T
C
2. Using your compass and straightedge, construct and label the centroid Q (the
intersection of the three medians) of ΔABC.
A
C
B
3. Using your compass and straightedge, construct and label the centroid L of ΔXYZ.
Y
Z
L
X
4. Using your compass and straightedge, construct AB
GH at B.
A
G
H
onstruct
An altitude of a triangle is the perpendicular segment from a vertex to the
opposite side or to the line that contains the opposite side.
Q
P
altitude from
Q to PR
R
Q
P
R
5. Using your compass and straightedge, construct two altitudes of the given triangle.
Label O, the point of intersection of the altitudes.
• The three altitudes of a triangle are concurrent.
• The point of concurrency is called the ORTHOCENTER.
• The orthocenter can be located inside, outside or on the triangle.
6.
Construct and label O, the orthocenter of ΔEFG.
Let's Sum it Up!!
How to construct the centroid: http://www.mathopenref.com/constcentroid.html
How to construct the orthocenter:
Example
http://www.mathopenref.com/constorthocenter.html
Point of
Concurrency
Property
Example
perpendicular
bisector
The circumcenter P of a triangle
circumcenter is equidistant from the vertices
of the triangle.
angle
bisector
incenter
The incenter I of a triangle is
equidistant form the sides of the
triangle.
centroid
The centroid R of a triangle is
two-thirds of the distance from
each vertex to the midpoint of the
opposite side.
median
altitude
orthocenter
The lines containing the altitudes
of a triangle are concurrent at
the orthocenter O.
P
I
R x
2x
3x
O
Name ______________________
Date ________________
CC Geometry H
HW #11 Centroid and Orthocenter
1) In ΔLMN, MC is the median from M to side LN. H is the centroid of ΔLMN and
mMH = 7.
L
C
H
Find mHC = _____ and mMC = ____
M
N
2) In ΔABC. E is the midpoint of AB, D is the midpoint of BC, and F is the midpoint
of AC. G is the centroid.
a) If the length of AD is 30, find the length of AG.
b) If the length of BG is 18, find the length of GF.
c) If the length of EG is 8, find the length of CE.
d) If the length of AG is 5x + 1, and the length of GD is x + 7, find the length of AD.
e) If the length of EG is x + 6, and the length of CE is x2, find the length of CE.
3) Using a compass and straightedge, construct the centroid of ΔRST.
R
S
T
4) Construct LT
5) Locate the orthocenter of ΔDEF.
XY at T.
E
Y
X
D
F
L
Y
6) Construct L, the orthocenter of ΔXYZ.
X
Z
Review:
1) If k ll m and n ll p, and m≮1 = 3x + 10 and
m≮2 = x + 90, find x and m≮1.
n
p
2
1
2) If QT
PR, find x and y.
P
k
m
3) Find the value of x.
State a reason for your equation.
(9x+16)0
300
T
(6x+15)0
(19x+3)
0
400
Q
y
x
R