Warm-Up

1. Warm-Up Three or more lines that intersect at the same point are called concurrent lines. The point of intersection is called the point of concurrency.

2. Example 1 Are the lines represented by the equations below concurrent? If so, find the point of concurrency. x+ y = 7 x+ 2y = 10 x- y = 1 Pick 2 equations and solve them for x & y Plug the values into all 3 equations and see if they make true statements x=4 y=3 Yes

3. 5.2-5.4: Points of Concurrency Objectives: • To define various points of concurrency • To discover, use, and prove various theorems about points of concurrency

4. Intersecting Medians Activity The centroid of a triangle divides each median into two parts. Clickthe button below to investigate the relationship of the 2 parts.

5. Concurrency of Medians Theorem 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.

6. Centroid The three medians of a triangle are concurrent. The point of concurrency is an interior point called the centroid. It is the balancing point or center of gravity of the triangle.

7. Example 2 In ΔRST, Q is the centroid and SQ = 8. Find QW and SW. QW = 4 SQ = 12

8. Others Points of Concurrency Since a triangle has 3 sides, it seems obvious that a triangle should have 3 perpendicular bisectors, 3 angle bisectors, and 3 altitudes. But are these various segments concurrent?

9. Others Points of Concurrency In this activity, we will use patty paper to investigate other possible points of concurrency, and then, hopefully, something magical will happen…

10. Circumcenter Concurrency of Perpendicular Bisectors of a Triangle Theorem The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.

11. Circumcenter The point of concurrency of the three perpendicular bisectors of a triangle is called the circumcenterof the triangle. In each diagram, the circle circumscribesthe triangle.

12. Explore Explore the perpendicular bisectors of a triangle and its circumcenterby clicking the button below

13. Incenter Concurrency of Angle Bisectors of a Triangle Theorem The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle.

14. Incenter The point of concurrency of the three angle bisectors of a triangle is called the incenterof the triangle. In the diagram, the circle is inscribedwithin the triangle.

15. Explore Explore the angle bisectors of a triangle and its incenterby clicking the button below

16. Orthocenter Concurrency of Altitudes of a Triangle Theorem The lines containing the altitudes of a triangle are concurrent. G

17. Orthocenter The point of concurrency of all three altitudes of a triangle is called the orthocenterof the triangle. The orthocenter, P, can be inside, on, or outside of a triangle depending on whether it is acute, right, or obtuse, respectively.

18. Explore • Explore the altitudes of a triangle and its orthocenter by clicking the button below.

19. Example 3 Is it possible for any of the points of concurrency to coincide? In other words, is there a triangle for which any of the points of concurrency are the same. Record your thoughts/predictions in your notebook

20. Example 4 Is it possible for any of the points of concurrency to be collinear?

21. Euler Line The Euler Line is the line that contains the orthocenter, centroid, and the circumcenter of a triangle.

22. Explore Click the button below to explore the Euler Line

23. Calculate in your notebook

24. Calculate in your notebook