2. Definition of a
Median of a Triangle
A median of a triangle is a segment whose endpoints are a vertex and a midpoint of the opposite side

3. Just to make sure we are clear about what an opposite side is…..
Given ABC, identify the opposite side
of A.
of B.
of C. BC AC AB

4. Any triangle has three medians.B
Any triangle has three medians. L M A N C
Let L, M and N be the midpoints of AB, BC and AC respectively.
CL, AM and NB are medians of ABC.

5. Where 3 or more lines intersect
A new term…
Point of concurrency
Where 3 or more lines intersect

6. Centroid Centroid The point where all 3 medians intersect
Is the point of
concurrency

7. 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

8. Centroid Theorem The centroid of a triangle divides the median into
segments with a 2:1 ratio.
The distance from the vertex to the centroid
is twice the distance
from the centroid to the midpoint.

9. VERTEX 3x 2x
CENTROID x
MIDPOINT

10. The distance from the vertex to the centroid is two-thirds
Centroid Theorem
The distance from the vertex
to the centroid is two-thirds
the distance from the
vertex to the midpoint

11. 𝟐 𝟑
QC = QZ
QC = 2CZ

12. PY = 21
PC =
CY = 8 12 4 8 8 6

13. Page 165 16 4

15. MID-SEGMENTS OF A TRIANGLE
A mid-segment of a triangle
connects the midpoints
of two sides of the triangle.

16. Mid-segment Theorem
The midsegment of a triangle is parallel to the third side and is half as long as that side. D B C E A

17. Identify the 3 pairs of parallel lines shown above

18. The mid-segment of a triangle is parallel to the third side and is half as long as that side.y y 2x x z z

19. 2b.
2a.

20. Example 1
In the diagram, ST and TU are midsegments of triangle PQR. Find PR and TU.
5 ft
16 ft
TU = ________
PR = ________

21. Example 3
In the diagram, ED and DF are midsegments of triangle ABC. Find DF and AB.
3X – 4
5X+2
2 (DF ) = AB
2 (3x – 4 ) = 5x + 2
6x – 8 = 5x + 2
x – 8 = 2
x = 10
x = ________ 10
DF = ________ 26
AB = ________ 52

22. ED is a mid-segment of ABC

27. Altitude .. Angle Bisector.. Perpendicular Bisector… Median ..
Quick notes
Angle Bisector..
Angle into 2 equal angles .. Incenter
Perpendicular Bisector…
90° .. bisects side .. Circumcenter
Median ..
Vertex .. Midpoint of side ..Centroid
Altitude ..
Vertex .. 90° .. Orthocenter