1. Warm-Up  The perpendicular bisectors meet at G. If BD = 4 and GD = 3, what is the length of GC?

2. Properties of Triangles – Day 3 Medians and Altitudes

3. Using the triangle and a pencil….  Can you balance the triangle on the tip of the pencil? Do you think every triangle has a balancing point?

4. How to find the balancing point…  Use your ruler to find the midpoint of each side and put a point there.  Draw a line connected the midpoint to the opposite vertex of the triangle.  The point where the lines meet is the balancing point of the triangle.

5. Theorem: Concurrency of Medians  The centroid is 2/3 the distance from each vertex to the midpoint of the opposite side.

6. Example 1: D is the centroid of the triangle and BE is perpendicular to AC.

7. Example 2:  Draw a triangle with vertices: D(3,6), F(7,10), and E(5,2)  Find the midpoint of each side  Find the centroid P

8. Theorem: Concurrency of Altitudes  The lines containing the altitudes intersect at a point called the orthocenter.

9. Name the line segment described:  Q:Perpendicular segment from vertex to opposite side.  A: Altitude  Q: Segment that divides an angle of a triangle into two congruent, adjacent angles.  A: Angle Bisector  Q: Perpendicular segment that intersects the side of a triangle at its midpoint.  A: Perpendicular Bisector  Q: Segment connecting a vertex of a triangle to the midpoint of the opposite side.  A: Median  Q: Segment that connects two midpoints of a triangle.  A: Midsegment

10. Name the concurrent points for the following segments:  Q: Angle Bisectors  A: Incenter  Q: Medians  A: Centroid  Q: Perpendicular Bisectors  A: Circumcenter  Q: Altitudes  A: Orthocenter

11. Homework – Day 3

12. Days 1 – 3 Review  Use your clickers to answer the following questions…

13. This segment’s endpoints are a vertex of a triangle and the midpoint of the opposite side. a.Median b.Perpendicular Bisector c.Midsegment d.Altitude

14. In  WXY, Q is the centroid and YQ = 2 x  15 and QA = 4. Find x. a.9.5 b.11.5 c.13.5 Q Y W X A B C

15. The circumcenter is equidistant to the _________ of a triangle. a.Vertices b.Sides

16. In  JKL, PS = 7. Find JP. a.7 b.14 c.21 J K L R S T P

17. This segment is perpendicular to a segment at its midpoint. a.Median b.Perpendicular Bisector c.Midsegment d.Altitude

18. This line passes through a vertex and divides that interior angle in half. a.Perpendicular Bisector b.Angle Bisector c.Midsegment

19. Find the measure of KF if K is the incenter of  ABC. a.5 b.12 c.13 A B C F D E 13 12 K

20. This is the intersection of the three perpendicular bisectors of a triangle and is equidistant from the vertices. a.Incenter b.Circumcenter c.Centroid d.Orthocenter

21. a. 2 b. 4 c. 6 D is the centroid of triangle ABC. Find CF.

22. This is the intersection of the three medians of a triangle and is 2/3 the distance from each vertex to the midpoint of the opposite side. a.Incenter b.Circumcenter c.Centroid d.Orthocenter

23. This is the intersection of the three angle bisectors of a triangle and is equidistant from the sides. a.Incenter b.Circumcenter c.Centroid d.Orthocenter

24. The incenter is equidistant to the _________ of a triangle. a.Vertices b.Sides

25. This is the intersection of the three altitudes of a triangle. a.Incenter b.Circumcenter c.Centroid d.Orthocenter

26. Find each measure of DC if D is the circumcenter of  ABC, AD = 12, and DF = 5. a.5 b.12 c.13 A B C D E F G

27. This is a perpendicular segment from a vertex to the opposite side. a.Median b.Perpendicular Bisector c.Midsegment d.Altitude