1. Medians of a Triangle
Section 4.6

2. Objective:
Identify and use medians in triangles.

3. Key Vocabulary Median of a triangle Concurrent lines
Point of concurrency
Centroid

4. Theorems
4.9 Centroid Theorem

5. 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

6. Median

7. 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. 7

8. Your Turn: Copy the triangle and draw a median. 1. 2. 3. ANSWER ANSWER
Sample answer: 1.
ANSWER
Sample answer: 2. 3.
ANSWER
Sample answer:

9. 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.

10. Definitions
Concurrent Lines – Three or more lines that intersect at a common point.
Point of Concurrency – The point where concurrent lines intersect.

11. 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.

12. 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.

13. CENTROID Intersection of all 3 medians
ALWAYS INSIDE THE TRIANGLE
The centroid is the point of balance for a ∆.

14. 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

15. Centroid
Distance from vertex to centroid is twice the distance from centroid to midpoint. 2x x
Vertex to Centroid  LONGER
Centroid to Midpoint  shorter
x + 2x = whole median

16. 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.
Short side of median
(midpoint to centroid)
is1/3 length of median
Long side of median
(vertex to centroid) is
2/3 length of median

17. Centroid 32 16 X 10 2x 5 x The centroid is 2/3’s of the distance
from the vertex to the side.

18. Example 2: 13 A B F X E C D
Find CX?
CX = 2(XF)
CX = 2(13)
CX = 26

19. Example 3: 18 A B F X E C D
Find XD?
AX = 2(XD)
18 = 2(XD)
9 = XD

20. In ABC, AN, BP, and CM are medians.
Example 4:
In ABC, AN, BP, and CM are medians. C
If EM = 3, find EC. N P E
EC = 2(3) B
EC = 6 M A

21. In ABC, AN, BP, and CM are medians.
Example 5: C
In ABC, AN, BP, and CM are medians. N
If EN = 12, find AN. P E B
AE = 2(12)=24 M
AN = AE + EN A
AN =
AN = 36

22. 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.

23. Problems

24. Example 6 In ΔXYZ, P is the centroid and YV = 12. Find YP and PV.
Centroid Theorem
YV = 12
Simplify.

25. Example 6 YP + PV = YV Segment Addition 8 + PV = 12 YP = 8
PV = 4 Subtract 8 from each side.
Answer: YP = 8; PV = 4

26. Example 7: Using the Centroid of a Triangle
P is the centroid of ∆QRS shown and PT = 5. Find RT and RP.

27. Example 7: Solution Because P is the centroid. RP = 2/3 RT.
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.

28. 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

29. Example 8
In ΔABC, CG = 4. Find GE.

30. Example 8 Centroid Theorem Segment Addition and Substitution CG = 4
Distributive Property

31. Example 8
Subtract GE from each side. __ 1 3
Answer: GE = 2

32. Your Turn
In ΔJLN, JP = 16. Find PM.
A. 4
B. 6
C. 16
D. 8

33. Example 9 EA has a length of 18 and DE has a length of 9.
E is the centroid of ∆ABC and DA = 27. Find EA and DE.
SOLUTION
Using Theorem 4.9, you know that
EA =
DA =
(27) = 18. 2 3
Now use the Segment Addition Postulate to find ED.
DA = DE + EA
Segment Addition Postulate
27 = DE + 18
Substitute 27 for DA and 18 for EA.
27 – 18 = DE + 18 – 18
Subtract 18 from each side.
9 = DE
Simplify.
ANSWER
EA has a length of 18 and DE has a length of 9. 33

34. Example 10 The median RT has a length of 15.
P is the centroid of ∆QRS and RP = 10. Find the length of RT.
SOLUTION
Use Theorem 4.9. 2 3
RP = RT 2 3
10 = RT
Substitute 10 for RP. = 3 2
(10) RT
Multiply each side by .
Simplify.
15 = RT
ANSWER
The median RT has a length of 15. 34

35. Your Turn: The centroid of the triangle is shown. Find the lengths.4.
Find BE and ED, given BD = 24.
ANSWER
BE = 16; ED = 8 5.
Find JG and KG, given JK = 4.
ANSWER
JG = 12; KG = 8 6.
Find PQ and PN, given QN = 20.
ANSWER
PQ = 10; PN = 30

36. Example 11: Finding the Centroid on Coordinate Plane
Find the coordinates of the centroid of ∆JKL
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.

37. Example 11: Finding the Centroid of a Triangle
The coordinates of N are:
3+7 , 6+10 = 10 , 16
Or (5, 8)
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.

38. Example 11: Finding the Centroid of a Triangle
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).

39. 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.

40. 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.

41. Example 12
Find the midpoint D of side BC.
Graph point D.

42. Example 12
Notice that is a horizontal line. The distance from D(3, 4 ) to A(1, 4) is 3 – 1 or 2 units.

43. Example 12
The centroid is the distance. So, the centroid is (2) or units to the right of A. The coordinates are

44. Example 12 Answer: The artist should place the pole at the point
Check Check the distance of the centroid from point D (3, 4). The centroid should be (2) or units to the left of D. So, the coordinates of the centroid are __ 1 3 2

45. Your Turn 5 A. (– , 2) B. (– , 2) 7 C. (–1, 2) 3 D. (0, 4) __
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?
A. (– , 2)
B. (– , 2)
C. (–1, 2)
D. (0, 4) __ 7 3 5

46. Assignment
(Finally!)
Pg. 209 – 211 #1 – 17 odd, 19 – 21 all, 23 – 29 odd