1. Assignment P. 307-309: 16, 17, 25, 28a, 30 P. 314-315: 19-25, 29 P. 322-323: 3, 5, 7, 8, 16, 33, 35, 36 Challenge Problems Print Triangle Vocab WS

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

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

4. 5.2-5.4: Points of Concurrency Objectives: 1.To define various points of concurrency 2.To discover, use, and prove various theorems about points of concurrency

5. Intersecting Medians Activity In this activity, we are going to find the balancing point for a given triangle. 1.Draw a triangle on a sturdy piece of paper (or wood), then cut it out.

6. Intersecting Medians Activity 2.Balance the triangle on the eraser end of a pencil. 3.Mark the balancing point on your triangle.

7. Intersecting Medians Activity 4.Use a ruler to find the midpoint of each side of the triangle. 5.Draw the three medians of the triangle.

8. Intersecting Medians Activity centroid 6.What do you notice about the point of concurrency of the three medians and the balancing point of the triangle? This point is called the centroid of the triangle.

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

10. 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.

11. Centroid 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.

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

13. 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?

14. 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…

15. Other Points of Concurrency 1.You’ll need 5 pieces of patty paper. On one piece, use your straightedge to draw a large acute or right triangle. 2.Trace the triangle you just made onto your 4 other pieces of patty paper. The best way to do this is to simply copy the vertices, and then connect the dots with your straightedge.

16. Other Points of Concurrency 3.Label each patty paper with one of the following: Medians, Perpendicular Bisectors, Angle Bisectors, Altitudes, and Surprise.

17. Medians 1.Find the midpoint of each side of your triangle by folding and pinching. 2.Connect each midpoint to its opposite vertex. centroid 3.We know that medians are concurrent at the centroid, so label this point G. 4.Take a second to admire your work, and then set it aside.

18. Perpendicular Bisectors 1.Find the perpendicular bisectors of each side of your triangle by folding and creasing. circumcenter 2.Are these lines concurrent? If so, label the point of concurrency C for circumcenter. 3.Since the circumcenter is on each of the perpendicular bisectors, what must be true about it?

19. Perpendicular Bisectors 6.With your compass, set the radius from the circumcenter to one of the vertices of your triangle, and then draw a circle. 7.Magic.

20. 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.

21. Circumcenter circumcenter The point of concurrency of the three perpendicular bisectors of a triangle is called the circumcenter of the triangle. circumscribes In each diagram, the circle circumscribes the triangle.

22. Explore Explore the perpendicular bisectors of a triangle and its circumcenter by clicking the button below

23. Angle Bisectors 1.Bisect each angle of your triangle by folding and creasing. incenter 2.Are the angle bisectors concurrent? If so, label the point of concurrency I for incenter. 3.Since the incenter is on each of the angle bisectors, what must be true about it?

24. Angle Bisectors 5.Find the perpendicular distance to one side of your triangle by folding and creasing. The fold should pass through I, and the side of the triangle should fold on top of itself. Label the point where the crease intersects the triangle point A. 6.Stretch your compass from I to A, and then draw a circle. 7.More magic.

25. 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.

26. Incenter incenter The point of concurrency of the three angle bisectors of a triangle is called the incenter of the triangle. inscribed In the diagram, the circle is inscribed within the triangle.

27. Explore Explore the angle bisectors of a triangle and its incenter by clicking the button below

28. Altitudes 1.Find the altitude to each side of your triangle. To do this, fold the patty paper so that the crease passes through a vertex and so that the opposite side folds on top of itself. orthocenter 2.Are the altitudes concurrent? If so, label the point of concurrency O for orthocenter.

29. Altitudes 3.There’s nothing terribly interesting about the orthocenter. 4.Or is there?

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

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

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

33. 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

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

35. Surprise 1.Lay the surprise patty paper on top of the triangle containing the medians. Copy the centroid. 2.Repeat for the circumcenter, incenter, and orthocenter. Euler Line 3.Are any of these points collinear? Obviously any two points would be collinear, but how about more? If so, use your straightedge to draw the line and label it Euler Line.

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

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

38. Secret Surprise 1.Bisect the segment OC on the Euler Line. Label this point with the number 9. 2.Copy point 9 onto the triangle containing the medians. 3.Stretch the compass from point 9, the center, to one of the midpoints, and then draw the circle. 4.More magic than you can handle.

39. Secret Surprise nine-point center nine-point circle 5.The point 9 is called the nine-point center, and the circle you just made is called the nine-point circle because it passes through 9 interesting points associated with the triangle. The three midpoints are 3 of the points. For ten bonus points on the next quiz, find and label the other six points.

40. Calculate in your notebook

42. Assignment P. 307-309: 16, 17, 25, 28a, 30 P. 314-315: 19-25, 29 P. 322-323: 3, 5, 7, 8, 16, 33, 35, 36 Challenge Problems Print Triangle Vocab WS