1. 5.4 Medians and Altitudes
Say what?????

2. Vocabulary…
Concurrent- 3 or more lines, rays, or segments that intersect at the same point
Median of a Triangle – a segment from a vertex to the midpoint of the opposite side
Centroid – point of concurrency of 3 medians
Altitude of a Triangle – the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side – there are 3 in a

3. Orthocenter - the point where the 3 altitudes of a intersect
Theorem 5.8: Concurrency of Medians
of a
-the medians of a intersect at a point that is 2/3 the distance from each vertex to the midpoint of the opposite side.
Theorem 5.9: Concurrency of Altitudes
-the lines containing the altitudes of a
are concurrent

4. In RST, Q is the centroid and SQ = 8. Find QW and SW.2 3 SW
Concurrency of Medians of a Triangle Theorem
8 = 2 3 SW
Substitute 8 for SQ.
Multiply each side by the reciprocal, . 3 2
12 = SW
Then QW =
SW – SQ =
12 – 8 = 4.
So, QW = 4 and SW = 12.

5. With a partner, do #’s 1-2 on p.278

6. With a partner, do #’s 3-4 on p.279

7. Practice In Exercises 1–3, use the diagram. G is the centroid of ABC.
1. If BG = 9, find BF.
ANSWER
13.5
2. If BD = 12, find AD.
ANSWER 12
3. If CD = 27, find GC.
ANSWER 18

8. Find the orthocenter.
Find the orthocenter P in an acute, a right, and an obtuse triangle. (Draw 3 altitudes…drop perpendicular lines from vertex to opposite side.)
SOLUTION
Acute triangle
P is inside triangle.
Right triangle
P is on triangle.
Obtuse triangle
P is outside triangle.

9. Can you answer these?????? Look at the and answer the following:
1. Is BD a median of ABC?
2. Is BD an altitude ABC?
3. Is BD a perpendicular bisector? B B
A D C A D C