2. 5.4Use Medians and Altitudes Theorem 5.8: Concurrency of Medians of a Triangle The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side. B A C D P F E

3. 5.4Use Medians and Altitudes Example 1 Use the centroid of a triangle In FGH, M is the centroid and GM = 6. Find ML and GL. G F H J M L K 6 Concurrency of Medians of a Triangle Theorem _____ = ____ GL Substitute ___ for GM. ___ = ____ GL Multiple each side by the reciprocal, ___. ___ = GL Then ML = GL – ____ = ___ – ____ = ___. So, ML = ___ and GL = ___.

4. 5.4Use Medians and Altitudes Checkpoint. Complete the following exercises. 1.In Example 1, suppose FM = 10. Find MK and FK. G F H J M L K 10

5. 5.4Use Medians and Altitudes The vertices of JKL are J(1, 2), K(4, 6), and L(7, 4). Find the coordinates of the centroid P of JKL. Sketch JKL. Then use the Midpoint Formula to find the midpoint M of JL and sketch median KM. Example 2 Find the centroid of a triangle J K L M The centroid is _________ of the distance from each vertex to the midpoint of the opposite side. two thirds The distance from vertex K to point M is 6 – ___ = ___ units. 3 3 So, the centroid is ___ (___) = ___ units down from K on KM. The coordinates of the centroid P are (4, 6 – ___), or (____). P

6. 5.4Use Medians and Altitudes Theorem 5.9: Concurrency of Altitudes of a Triangle The lines containing the altitudes of a triangle are ___________. The lines containing AF, BE, and CD meet at G C B A G E D F

7. 5.4Use Medians and Altitudes Find the orthocenter P of the triangle. Example 3 Find the orthocenter a. b. Solution a. b. P P

8. 5.4Use Medians and Altitudes Checkpoint. Complete the following exercises. 2.In Example 2, where do you need to move point K so that the centroid is P(4, 5)? J K L P M Distance from the midpoint to the centroid is how much of the total distance of the median? If that distance is 2, what is the total distance?

9. 5.4Use Medians and Altitudes Checkpoint. Complete the following exercises. 3.Find the orthocenter P of the triangle. P

10. 5.4Use Medians and Altitudes Pg. 294, 5.4 #1-19