5.4

1. 5.4 Use Medians and Altitudes B E D C A F 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. P

2. 5.4 Use Medians and Altitudes In FGH, M is the centroid and GM = 6. Find ML and GL. G M K J H F L Use the centroid of a triangle Example 1 6 _____ = ____GL Concurrency of Medians of a Triangle Theorem ___ = ____GL Substitute ___ for GM. ___ = GL Multiple each side by the reciprocal, ___. Then ML = GL – ____ = ___ – ____ = ___. So, ML = ___ and GL = ___.

3. 5.4 Use Medians and Altitudes G M K J H F L Checkpoint. Complete the following exercises. • In Example 1, suppose FM = 10. Find MK and FK. 10

4. 5.4 Use 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. Find the centroid of a triangle Example 2 K L The centroid is _________ of the distance from each vertex to the midpoint of the opposite side. two thirds P M J 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 (____).

5. 5.4 Use Medians and Altitudes A B C Theorem 5.9: Concurrency of Altitudes of a Triangle The lines containing the altitudes of a triangle are ___________. G E D The lines containing AF, BE, and CD meet at G F

6. 5.4 Use Medians and Altitudes Find the orthocenter Example 3 Find the orthocenter P of the triangle. Solution P P

7. 5.4 Use Medians and Altitudes Checkpoint. Complete the following exercises. • In Example 2, where do you need to move point K so that the centroid is P(4, 5)? Distance from the midpoint to the centroid is how much of the total distance of the median? K If that distance is 2, what is the total distance? P L M J

8. 5.4 Use Medians and Altitudes Checkpoint. Complete the following exercises. • Find the orthocenter P of the triangle. P

9. 5.4 Use Medians and Altitudes Pg. 294, 5.4 #1-19