Aim #33: What are medians of triangles?
Do Now: In ΔABC below, D is the midpoint of AB, E is the midpoint of BC, and F
is the midpoint of AC. Complete each statement:
DE is || to _____ and measures ____ the length of _____.
DF is || to _____ and measures ____ the length of _____.
EF is || to _____ and measures ____ the length of _____.
The vertices of ΔXYZ are X(4,5), Y(6,1) and Z(8,9).
a) Graph and label ΔXYZ.
b) Determine the midpoints of XY,
YZ and XZ. Label them E, and F,
and G, respectively.
c) Connect each vertex to the
midpoint of the opposite side.
Label the intersection of these
three segments as C.
What is this point called? ___________
d) Verify that C divides the medians
into a 2:1 ratio.
XC = ___ CF = ____
YC = ___ CG = ____
ZC = ___ CE = ____
The median of a triangle is a segment that connects each vertex to the midpoint of the
opposite side. For example, in triangle ABC below, AM is a median.
B
M
C
A
The three medians of a triangle are concurrent (they meet at a single point). This
point of concurrency is called the
. This point is the center of gravity
of the triangle.
There is a length relationship for each median: The length from the __________
to the ______________ is twice the length from the ___________ to the
____________________________.
1) In the figure, DF = 4, BF = 16, and AG = 30. Find the
following:
a) FC =
d) BE =
b) DC =
e) FG =
c) AF =
f) EF =
2) G is the centroid of triangle ABC. BG = 6, AF = 12, and AE = 15. Find the
length of the following segments:
B
a) FC =
b) BF =
D
E
G
c) AG =
d) GE =
A
C
F
3) In ΔABC, E is the midpoint of AB, D is the midpoint of BC, and F is the midpoint
B
of AC.
a) If the length of AD is 30, find AG.
E
G
D
b) If the length of BG is 18, find GF.
A
F
C
c) If the length of EG is 8, find CE.
d) If the length of AG is 5x + 2, and the length of GD is x + 7, find AD.
e) If the length of EG is x + 6, and the length of CE is x2, find CE.
Given: ΔABC with D, E, and F the midpoints of sides AB, BC, and
AC, respectively. H is midpoint of AG and J is midpoint of GC.
Prove: The distance from a vertex to the centroid
equals twice the distance from the centroid to
midpoint.
AG = 2GE, CG = 2GD
Statements
Reasons
1. DE || AC
2. HJ || AC
3. DE || HJ
4. DE = 1 /2 AC and HJ = 1 /2 AC
5. DE = HJ
6. DEJH is a parallelogram.
7. DJ and HE bisect each other
8. HG = GE, JG = GD
9. AH = HG, CJ = JG
10. AH = HG = GE and CJ = JG = GD
11. AG = AH + HG, CG = CJ + JG
12. AG = GE + GE or 2GE
CG = GD + GD or 2GD
If a median were drawn from B to AC, what point would it pass through in the
diagram above? ______
The three medians above are concurrent at point ____, also known as the centroid
of ΔABC.
Construction
Ty is building a model of a hang glider using the template below. To place his
supports accurately, Ty needs to locate the center of gravity on his model.
a. Use your compass and straightedge to locate the center of gravity on Ty‛s model.
b. What is another name for the center of gravity?
c. Describe the relationship between the longer and shorter sections of the line
segments you drew as you located the center of gravity.
Let's Sum It Up!!
The three medians of a triangle are concurrent at the centroid, or the center of
gravity. This point of concurrency divides the length of each median in a ratio of 2:1;
the length from the vertex to the centroid is twice the length from the centroid to
the midpoint of the other side.
Name ___________________
Common Core Geometry H
Date _____________________
HW #33 (Medians)
1. DQ, FP, and RE are all medians of ΔDEF. DQ = 24, FC = 10, and RC = 7. Find:
a. DC =
b. CQ =
c. FP =
d. CE =
2. In ΔRST, C, K, and M are midpoints of sides RS, ST, and TR respectively.
The perimeter of ΔMCK is 54 units and the ratio of the lengths of its sides is
given by CK : KM : MC = 4 : 3 : 2.
a. Find the measure of CK, KM, and MC.
b. Find the perimeter of ΔRST.
3. In ΔABC, medians EC, FB and DA meet at G.
BG = x2 - 5x - 28, GF = x - 5, BE = 2y + 3, EA = 5y - 3, AD = 2z + 1, and DG = z - 1.
Find the length of AB, BG, and AD.
A
E
G
B
D
F
C
4. In the figure, ΔABC is reflected over
AB to create ΔABD. Points P, E, and F are
midpoints of AB, BD, and BC, respectively.
If AH = AG, prove that PH = GP.
* Rigid motions may be used as a reason in this proof.* Statements
1. AF = AE
Reasons
1.
2. ≮FAB ≅ ≮EAB
2.
3. AH = AG
3.
4. AP = AP
4.
5. ΔAPH ≅ ΔAPG
5.
6. PH = GP
6.
Review:
1) The measures of two consecutive angles of a parallelogram are in the ratio 5:4.
What is the measure of an obtuse angle of the parallelogram?
2) Find the values of x and y.