Transcript Medians and Altitudes of Triangles

Medians of a Triangle
Section 4.6
Objective:

Identify and use medians in triangles.
Key Vocabulary




Median of a triangle
Concurrent lines
Point of concurrency
Centroid
Theorems

4.9 Centroid Theorem
Median of a Triangle


A median of a triangle is
a segments whose
endpoints are a vertex
of the triangle and the
midpoint of the opposite
side.
For instance in ∆ABC,
shown at the right, D is
the midpoint of side BC.
So, AD is a median of
the triangle
A
MEDIAN
C
D
B
Median
Example 1
In ∆STR, draw a median from S to its
opposite side.
SOLUTION
The side opposite S is TR.
Find the midpoint of TR, and label it P.
Then draw a segment from point S to
point P. SP is a median of ∆STR.
Your Turn:
Copy the triangle and draw a median.
1.
ANSWER
Sample answer:
ANSWER
Sample answer:
ANSWER
Sample answer:
2.
3.
Concurrent Lines
When two lines intersect at one point, we say that the lines
are intersecting. The point at which they intersect is
the point of intersection.
(nothing new right?)
Well, if three or more lines intersect at a
common point, we say that the lines are
concurrent lines. The point at which
these lines intersect is called the point
of concurrency.
Definitions


Concurrent Lines – Three or more lines that
intersect at a common point.
Point of Concurrency – The point where
concurrent lines intersect.
Medians



Every triangle has three
medians that are
concurrent.
The point of
concurrency of the
medians of a triangle is
called the CENTROID.
The centroid, labeled P
in the diagram is
ALWAYS inside the
triangle.
Centroid
The centroid is the center of balance for the triangle. You can
balance a triangle on the tip of
your pencil if you place the tip on
the centroid
Finding balancing points of objects is important in
engineering, construction, and science.
CENTROID
CENTROID
P
P
centroid
P
RIGHT TRIANGLE
obtuse triangle
acute triangle
Intersection of all 3 medians
ALWAYS INSIDE THE TRIANGLE
The centroid is the point of balance for a ∆.
centroid
Theorem 4.9
Centroid Theorem


The medians of a triangle
intersect at a point called
the centroid that is two
thirds of the distance from
each vertex to the midpoint
of the opposite side.
Example: If P is the
centroid of ∆ABC, then
AP = 2/3 AD,
BP = 2/3 BF, and
CP = 2/3 CE
B
D
E
C
P
F
A
Centroid
Distance from vertex to
centroid is twice the
distance from centroid to
midpoint.
Vertex to Centroid  LONGER
Centroid to Midpoint  shorter
x + 2x = whole median
Centroid
So, if you know the length of any median, you know where the
three medians are concurrent. It would be at the point that is
2/3 the length of the median from the vertex it originated from.
Long side of median
(vertex to centroid) is
2/3 length of median
Short side of median
(midpoint to centroid)
is1/3 length of median
Centroid
10
2x 32
5x
16
X
The centroid is 2/3’s of the distance
from the vertex to the side.
Example 2:
C
CX = 2(XF)
D
E
CX = 2(13)
CX = 26
X
A
F
B
Example 3:
C
AX = 2(XD)
18 = 2(XD)
D
E
9 = XD
X
A
F
B
Example 4:
In ABC, AN, BP, and CM are medians.
C
If EM = 3, find EC.
N
P
E
EC = 2(3)
EC = 6
B
A
M
Example 5:
C
In ABC, AN, BP, and CM are medians.
If EN = 12, find AN.
P
N
E
B
AE = 2(12)=24
AN = AE + EN
AN = 24 + 12
AN = 36
A
M
CENTROID

Facts about Medians and the Centroid




Medians connect a vertex and the
midpoint of the opposite side.
The point of concurrency of the medians
is called the centroid.
The centroid is the balancing point of a
triangle.
The centroid is two-thirds of the distance
from each vertex to the midpoint of the
opposite side.
Problems
Example 6
In ΔXYZ, P is the centroid and
YV = 12. Find YP and PV.
Centroid Theorem
YV = 12
Simplify.
Example 6
YP + PV = YV
8 + PV = 12
PV = 4
Answer: YP = 8; PV = 4
Segment Addition
YP = 8
Subtract 8 from each side.
Example 7: Using the Centroid of a
Triangle
R
P is the centroid of
∆QRS shown and
PT = 5. Find RT
and RP.
S
P
T
Q
Example 7: Solution
Because P is the centroid.
RP = 2/3 RT.
R
Then PT= RT – RP = 1/3
RT. Substituting 5 for
PT, 5 = 1/3 RT, so
RT = 15.
Then RP = 2/3 RT =
2/3 (15) = 10
► So, RP = 10, and RT =
15.
S
P
T
Q
Your Turn
In ΔLNP, R is the centroid and
LO = 30. Find LR and RO.
A. LR = 15; RO = 15
B. LR = 20; RO = 10
C. LR = 17; RO = 13
D. LR = 18; RO = 12
Example 8
In ΔABC, CG = 4. Find GE.
Example 8
Centroid Theorem
Segment Addition and Substitution
CG = 4
Distributive Property
Example 8
1 GE from each side.
Subtract __
3
Answer: GE = 2
Your Turn
In ΔJLN, JP = 16. Find PM.
A. 4
B. 6
C. 16
D. 8
Example 9
E is the centroid of ∆ABC and DA = 27. Find EA and DE.
SOLUTION
Using Theorem 4.9, you know that
2
2
EA =
DA =
(27) = 18.
3
3
Now use the Segment Addition Postulate to find ED.
DA = DE + EA
27 = DE + 18
27 – 18 = DE + 18 – 18
9 = DE
ANSWER
Segment Addition Postulate
Substitute 27 for DA and 18 for EA.
Subtract 18 from each side.
Simplify.
EA has a length of 18 and DE has a length
of 9.
Example 10
P is the centroid of ∆QRS and RP = 10. Find the length
of RT.
SOLUTION
2
RP =
RT
3
10 =
2
RT
3
3
3
(10) =
2
2
15 = RT
ANSWER
2
RT
3
Use Theorem 4.9.
Substitute 10 for RP.
Multiply each side by 3 .
2
Simplify.
The median RT has a length of 15.
Your Turn:
The centroid of the triangle is shown. Find the
lengths.
4.
Find BE and ED, given BD = 24.
ANSWER
5.
Find JG and KG, given JK = 4.
ANSWER
6.
BE = 16; ED = 8
JG = 12; KG = 8
Find PQ and PN, given QN = 20.
ANSWER
PQ = 10; PN = 30
Example 11: Finding the Centroid on
Coordinate Plane
J (7, 10)
10
Find the coordinates of the
centroid of ∆JKL
N
8
6
L (3, 6)
P
4
M
2
K (5, 2)
You know that the centroid
is two thirds of the
distance from each
vertex to the midpoint of
the opposite side.
Choose the median KN.
Find the coordinates of
N, the midpoint of JL.
Example 11: Finding the Centroid of a
Triangle
J (7, 10)
10
N
The coordinates of N are:
3+7 , 6+10 = 10 , 16
2
2
2 2
8
Or (5, 8)
6
L (3, 6)
P
4
M
2
K (5, 2)
Find the distance from
vertex K to midpoint N.
The distance from K(5,
2) to N (5, 8) is 8-2 or 6
units.
Example 11: Finding the Centroid of a
Triangle
J (7, 10)
10
N
8
6
L (3, 6)
P
4
M
2
K (5, 2)
Determine the
coordinates of the
centroid, which is
2/3 ∙ 6 or 4 units up
from vertex K along
median KN.
►The coordinates of
centroid P are (5,
2+4), or (5, 6).
Example 12
SCULPTURE An artist is designing a sculpture that
balances a triangle on top of a pole. In the artist’s
design on the coordinate plane, the vertices are
located at (1, 4), (3, 0), and (3, 8). What are the
coordinates of the point where the artist should place
the pole under the triangle so that it will balance?
Understand
You need to find the centroid of the
triangle. This is the point at which the
triangle will balance.
Example 12
Plan
Graph and label the triangle with
vertices (1, 4), (3, 0), and (3, 8). Use
the Midpoint Theorem to find the
midpoint of one of the sides of the
triangle. The centroid is two-thirds the
distance from the opposite vertex to
that midpoint.
Solve
Graph ΔABC.
Example 12
Find the midpoint D of side BC.
Graph point D.
Example 12
Notice that
is a horizontal line.
The distance from D(3, 4 ) to A(1, 4)
is 3 – 1 or 2 units.
Example 12
The centroid is
the distance.
So, the centroid is
(2) or
units to
the right of A. The coordinates are
.
Example 12
Answer: The artist should place the pole at the point
Check
Check the distance of the centroid from
1 (2) or
point D (3, 4). The centroid should be __
3
__
2 units to the left of D. So, the
3
coordinates of the centroid are
.
Your Turn
BASEBALL A fan of a local baseball team is designing a
triangular sign for the upcoming game. In his design on the
coordinate plane, the vertices are located at (–3, 2), (–1, –2),
and (–1, 6). What are the coordinates of the point where the
fan should place the pole under the triangle so that it will
balance?
5 , 2)
A. (– __
3
7 , 2)
B. (– __
3
C. (–1, 2)
D. (0, 4)
Assignment
(Finally!)

Pg. 209 – 211 #1 – 17 odd, 19 – 21 all,
23 – 29 odd