1. Introduction Think about all the properties of triangles we have learned so far and all the constructions we are able to perform. What properties exist when the perpendicular bisectors of triangles are constructed? Is there anything special about where the angle bisectors of a triangle intersect? We know triangles have three altitudes, but can determining each one serve any other purpose? How can the midpoints of each side of a triangle help find the center of gravity of a triangle? Each of these questions will be answered as we explore the centers of triangles. 1 1.9.4: Proving Centers of Triangles

2. Key Concepts Every triangle has four centers. Each center is determined by a different point of concurrency—the point at which three or more lines intersect. These centers are the circumcenter, the incenter, the orthocenter, and the centroid. 2 1.9.4: Proving Centers of Triangles

3. Key Concepts, continued Circumcenters The perpendicular bisector is the line that is constructed through the midpoint of a segment. In the case of a triangle, the perpendicular bisectors are the midpoints of each of the sides. The three perpendicular bisectors of a triangle are concurrent, or intersect at one point. 3 1.9.4: Proving Centers of Triangles

4. Key Concepts, continued This point of concurrency is called the circumcenter of the triangle. The circumcenter of a triangle is equidistant, or the same distance, from the vertices of the triangle. This is known as the Circumcenter Theorem. 4 1.9.4: Proving Centers of Triangles

5. Key Concepts, continued 5 1.9.4: Proving Centers of Triangles Theorem Circumcenter Theorem The circumcenter of a triangle is equidistant from the vertices of a triangle. The circumcenter of this triangle is at X.

6. Key Concepts, continued The circumcenter can be inside the triangle, outside the triangle, or even on the triangle depending on the type of triangle. The circumcenter is inside acute triangles, outside obtuse triangles, and on the midpoint of the hypotenuse of right triangles. 6 1.9.4: Proving Centers of Triangles

7. Key Concepts, continued Look at the placement of the circumcenter, point X, in the following examples. 7 1.9.4: Proving Centers of Triangles Acute triangleObtuse triangleRight triangle X is inside the triangle. X is outside the triangle. X is on the midpoint of the hypotenuse.

8. 8 1.9.4: Proving Centers of Triangles Key Concepts, continued The circumcenter of a triangle is also the center of the circle that connects each of the vertices of a triangle. This is known as the circle that circumscribes the triangle.

9. 9 1.9.4: Proving Centers of Triangles Key Concepts, continued Incenters The angle bisectors of a triangle are rays that cut the measure of each vertex in half. The three angle bisectors of a triangle are also concurrent. This point of concurrency is called the incenter of the triangle. The incenter of a triangle is equidistant from the sides of the triangle. This is known as the Incenter Theorem.

10. Key Concepts, continued 10 1.9.4: Proving Centers of Triangles Theorem Incenter Theorem The incenter of a triangle is equidistant from the sides of a triangle. The incenter of this triangle is at X.

11. Key Concepts, continued The incenter is always inside the triangle. 11 1.9.4: Proving Centers of Triangles Acute triangleObtuse triangleRight triangle

12. Key Concepts, continued The incenter of a triangle is the center of the circle that connects each of the sides of a triangle. This is known as the circle that inscribes the triangle. 12 1.9.4: Proving Centers of Triangles

13. 13 1.9.4: Proving Centers of Triangles Key Concepts, continued Orthocenters The altitudes of a triangle are the perpendicular lines from each vertex of the triangle to its opposite side, also called the height of the triangle. The three altitudes of a triangle are also concurrent. This point of concurrency is called the orthocenter of the triangle.

14. 14 1.9.4: Proving Centers of Triangles Key Concepts, continued The orthocenter can be inside the triangle, outside the triangle, or even on the triangle depending on the type of triangle. The orthocenter is inside acute triangles, outside obtuse triangles, and at the vertex of the right angle of right triangles.

15. Key Concepts, continued Look at the placement of the orthocenter, point X, in the following examples. 15 1.9.4: Proving Centers of Triangles Acute triangleObtuse triangleRight triangle X is inside the triangle. X is outside the triangle. X is at the vertex of the right angle.

16. 16 1.9.4: Proving Centers of Triangles Key Concepts, continued Centroids The medians of a triangle are segments that join the vertices of the triangle to the midpoint of the opposite sides. Every triangle has three medians. The three medians of a triangle are also concurrent.

17. 17 1.9.4: Proving Centers of Triangles Key Concepts, continued This point of concurrency is called the centroid of the triangle. The centroid is always located inside the triangle the distance from each vertex to the midpoint of the opposite side. This is known as the Centroid Theorem.

18. Key Concepts, continued 18 1.9.4: Proving Centers of Triangles Theorem Centroid Theorem The centroid of a triangle is the distance from each vertex to the midpoint of the opposite side. The centroid of this triangle is at point X.

19. Key Concepts, continued The centroid is always located inside the triangle. The centroid is also called the center of gravity of a triangle because the triangle will always balance at this point. 19 1.9.4: Proving Centers of Triangles Acute triangleObtuse triangleRight triangle

20. Key Concepts, continued Each point of concurrency discussed is considered a center of the triangle. Each center serves its own purpose in design, planning, and construction. 20 1.9.4: Proving Centers of Triangles Center of triangleIntersection of… CircumcenterPerpendicular bisectors IncenterAngle bisectors OrthocenterAltitudes CentroidMedians

21. Common Errors/Misconceptions not recognizing that the circumcenter and orthocenter are outside of obtuse triangles incorrectly assuming that the perpendicular bisector of the side of a triangle will pass through the opposite vertex interchanging circumcenter, incenter, orthocenter, and centroid confusing medians with midsegments misidentifying the height of the triangle 21 1.9.4: Proving Centers of Triangles

22. Guided Practice Example 3 has vertices A (–2, 4), B (5, 4), and C (3, –2). Find the equation of each median of to verify that (2, 2) is the centroid of. 22 1.9.4: Proving Centers of Triangles

23. Guided Practice: Example 3, continued 1.Identify known information. has vertices A (–2,4), B (5, 4), and C (3, –2). The centroid is X (2, 2). The centroid of a triangle is the intersection of the medians of the triangle. 23 1.9.4: Proving Centers of Triangles

24. Guided Practice: Example 3, continued 2.Determine the midpoint of each side of the triangle. Use the midpoint formula to find the midpoint of. 24 1.9.4: Proving Centers of Triangles Midpoint formula Substitute (–2, 4) and (5, 4) for (x 1, y 1 ) and (x 2, y 2 ).

25. Guided Practice: Example 3, continued The midpoint of is. 25 1.9.4: Proving Centers of Triangles Simplify.

26. Guided Practice: Example 3, continued Use the midpoint formula to find the midpoint of. The midpoint of is (4, 1). 26 1.9.4: Proving Centers of Triangles Midpoint formula Substitute (5, 4) and (3, – 2) for (x 1, y 1 ) and (x 2, y 2 ). Simplify.

27. Guided Practice: Example 3, continued Use the midpoint formula to find the midpoint of. 27 1.9.4: Proving Centers of Triangles Midpoint formula Substitute ( – 2, 4) and (3, – 2) for (x 1, y 1 ) and (x 2, y 2 ).

28. Guided Practice: Example 3, continued The midpoint of is. 28 1.9.4: Proving Centers of Triangles Simplify.

29. Guided Practice: Example 3, continued 29 1.9.4: Proving Centers of Triangles

30. Guided Practice: Example 3, continued 3.Determine the medians of the triangle. Find the equation of, which is the line that passes through A and the midpoint of. Use the slope formula to calculate the slope of. 30 1.9.4: Proving Centers of Triangles Slope formula Substitute (–2, 4) and (4, 1) for (x 1, y 1 ) and (x 2, y 2 ).

31. Guided Practice: Example 3, continued The slope of is 31 1.9.4: Proving Centers of Triangles Simplify.

32. 32 1.9.4: Proving Centers of Triangles Guided Practice: Example 3, continued Find the y-intercept of. The equation of that passes through A and the midpoint of is. Point-slope form of a line Substitute (–2, 4) for (x 1, y 1 ) and for m. Simplify.

33. Guided Practice: Example 3, continued Find the equation of, which is the line that passes through B and the midpoint of. Use the slope formula to calculate the slope of. 33 1.9.4: Proving Centers of Triangles Slope formula Substitute (5, 4) and for (x 1, y 1 ) and (x 2, y 2 ).

34. Guided Practice: Example 3, continued The slope of is 34 1.9.4: Proving Centers of Triangles Simplify.

35. Guided Practice: Example 3, continued Find the y-intercept of. The equation of that passes through B and the midpoint of is. 35 1.9.4: Proving Centers of Triangles Point-slope form of a line Substitute (5, 4) for (x 1, y 1 ) and for m. Simplify.

36. Guided Practice: Example 3, continued Find the equation of, which is the line that passes through C and the midpoint of. Use the slope formula to calculate the slope of. 36 1.9.4: Proving Centers of Triangles Slope formula Substitute (3, –2) and for (x 1, y 1 ) and (x 2, y 2 ).

37. Guided Practice: Example 3, continued The slope of is 37 1.9.4: Proving Centers of Triangles Simplify.

38. Guided Practice: Example 3, continued Find the y-intercept of. The equation of that passes through C and the midpoint of is. 38 1.9.4: Proving Centers of Triangles Point-slope form of a line Substitute (3, –2) for (x 1, y 1 ) and –4 for m. Simplify.

39. Guided Practice: Example 3, continued 4.Verify that X (2, 2) is the intersection of the three medians. For (2, 2) to be the intersection of the three medians, the point must satisfy each of the equations: 39 1.9.4: Proving Centers of Triangles

40. Guided Practice: Example 3, continued (2, 2) satisfies the equation of the median from A to the midpoint of. 40 1.9.4: Proving Centers of Triangles Equation of the median from A to the midpoint of Substitute X (2, 2) for (x, y). Simplify.

41. Guided Practice: Example 3, continued (2, 2) satisfies the equation of the median from B to the midpoint of. 41 1.9.4: Proving Centers of Triangles Equation of the median from B to the midpoint of Substitute X (2, 2) for (x, y). Simplify.

42. Guided Practice: Example 3, continued (2, 2) satisfies the equation of the median from C to the midpoint of. 42 1.9.4: Proving Centers of Triangles Equation of the median from C to the midpoint of Substitute X (2, 2) for (x, y). Simplify.

43. Guided Practice: Example 3, continued 4.State your conclusion. X (2, 2) is the centroid of with vertices A (–2, 4), B (5, 4), and C (3, –2) because X satisfies each of the equations of the medians of the triangle. 43 1.9.4: Proving Centers of Triangles ✔

44. Guided Practice: Example 3, continued 44 1.9.4: Proving Centers of Triangles

45. Guided Practice Example 4 Using from Example 3, which has vertices A (–2, 4), B (5, 4), and C (3, –2), verify that the centroid, X (2, 2), is the distance from each vertex. 45 1.9.4: Proving Centers of Triangles

46. Guided Practice: Example 4, continued 1.Identify the known information. has vertices A (–2, 4), B (5, 4), and C (3, –2). The centroid is X (2, 2). The midpoints of are T, U (4, 1), and V. 46 1.9.4: Proving Centers of Triangles

47. Guided Practice: Example 4, continued 2.Use the distance formula to show that point X (2, 2) is the distance from each vertex. Use the distance formula to calculate the distance from A to U. 47 1.9.4: Proving Centers of Triangles Distance formula

48. Guided Practice: Example 4, continued The distance from A to U is units. 48 1.9.4: Proving Centers of Triangles Substitute (–2, 4) and (4, 1) for (x 1, y 1 ) and (x 2, y 2 ). Simplify.

49. Guided Practice: Example 4, continued Calculate the distance from X to A. 49 1.9.4: Proving Centers of Triangles Distance formula Substitute (2, 2) and (–2, 4) for (x 1, y 1 ) and (x 2, y 2 ).

50. Guided Practice: Example 4, continued The distance from X to A is units. 50 1.9.4: Proving Centers of Triangles Simplify.

51. Guided Practice: Example 4, continued X is the distance from A. 51 1.9.4: Proving Centers of Triangles Centroid Theorem Substitute the distances found for AU and XA. Simplify.

52. Guided Practice: Example 4, continued Use the distance formula to calculate the distance from B to V. 52 1.9.4: Proving Centers of Triangles Distance formula Substitute (5, 4) and for (x 1, y 1 ) and (x 2, y 2 ). Simplify.

53. Guided Practice: Example 4, continued The distance from B to V is units. 53 1.9.4: Proving Centers of Triangles

54. Guided Practice: Example 4, continued Calculate the distance from X to B. 54 1.9.4: Proving Centers of Triangles Distance formula Substitute (2, 2) and (5, 4) for (x 1, y 1 ) and (x 2, y 2 ).

55. Guided Practice: Example 4, continued The distance from X to B is units. 55 1.9.4: Proving Centers of Triangles Simplify.

56. Guided Practice: Example 4, continued X is the distance from B. 56 1.9.4: Proving Centers of Triangles Centroid Theorem Substitute the distances found for BV and XB. Simplify.

57. Guided Practice: Example 4, continued Use the distance formula to calculate the distance from C to T. 57 1.9.4: Proving Centers of Triangles Distance formula Substitute (3, –2) and for (x 1, y 1 ) and (x 2, y 2 ). Simplify.

58. Guided Practice: Example 4, continued The distance from C to T is units. 58 1.9.4: Proving Centers of Triangles

59. Guided Practice: Example 4, continued Calculate the distance from X to C. 59 1.9.4: Proving Centers of Triangles Distance formula Substitute (2, 2) and (3, –2) for (x 1, y 1 ) and (x 2, y 2 ).

60. Guided Practice: Example 4, continued The distance from X to C is units. 60 1.9.4: Proving Centers of Triangles Simplify.

61. Guided Practice: Example 4, continued X is the distance from C. 61 1.9.4: Proving Centers of Triangles Centroid Theorem Substitute the distances found for CT and XC. Simplify.

62. Guided Practice: Example 4, continued The centroid, X (2, 2), is the distance from each vertex. 62 1.9.4: Proving Centers of Triangles ✔

63. Guided Practice: Example 4, continued 63 1.9.4: Proving Centers of Triangles