1. Geometry
Unit 5: Triangle Parts

2. CONCURRENT:

3. Concurrent:
When three or more lines intersect at the same point, P P

4. MIDSEGMENT:

5. Segment connecting the midpoints of two sides of a triangle
Triangle Midsegment:
Segment connecting the midpoints of two sides of a triangle

6. Triangle Midsegments
Parallel to the third side of the triangle

7. Triangle Midsegments Parallel to the third side of the triangle
Half the length of the third side of the triangle

8. BISECTORS

9. Bisectors
Both types of bisectors (Angle Bisectors and Perpendicular Bisectors) will lead to circles.

10. The circles will be inscribed in or circumscribed about triangles.
Bisectors
Both types of bisectors (Angle Bisectors and Perpendicular Bisectors) will lead to circles.
The circles will be inscribed in or circumscribed about triangles.

11. PERPENDICULAR BISECTORS

12. Perpendicular Bisectors
Lines that bisect a side and are perpendicular to it

13. PERPENDICULAR BISECTOR
Bisects a side and makes a 90 angle with it

14. Perpendicular Bisectors (in purple)
Concurrent at the CIRCUMCENTER of each green triangle

15. Perpendicular Bisectors
Circumcenter can be inside, on, or outside of the triangle

16. Perpendicular Bisectors
Circle 1: circumcenter is outside triangle
Circle 2: circumcenter is inside triangle . .
circumcenter
circumcenter

17. Perpendicular Bisectors
purple radii of circle go from circumcenter to each vertex of the triangle r r

18. Perpendicular Bisectors
Can you identify three isosceles triangles in each figure? r r

19. Perpendicular Bisectors
Lines that bisect a side and are perpendicular to it
Concurrent at the Circumcenter of the triangle
Circumcenter can be inside, on, or outside of the triangle
Radii of circle go from circumcenter to each vertex of the triangle

20. Angle Bisector

21. Angle Bisector
Bisects the angle

22. Concurrent at the INCENTER (center of circle INSCRIBED in triangle)
Angle Bisector
Concurrent at the INCENTER
(center of circle INSCRIBED in triangle)

23. Concurrent at the INCENTER (center of circle INSCRIBED in triangle)
Angle Bisector
Concurrent at the INCENTER
(center of circle INSCRIBED in triangle)

24. MEDIAN

25. MEDIAN Segment from vertex to opposite side’s midpoint
(Nothing to do with the angle!!)

26. Concurrent at CENTROID (center of gravity)
MEDIAN
Concurrent at CENTROID
(center of gravity)

27. The center of gravity is the BALANCE POINT.
MEDIAN
The center of gravity is the BALANCE POINT.

28. The CENTROID must be inside of the triangle!
MEDIAN
The CENTROID must be inside of the triangle!

29. ALTITUDE

30. ALTITUDE
Perpendicular segment from a vertex to the side opposite (or extension of the side opposite).

31. Height of the triangle (perpendicular to the base)
ALTITUDE
Height of the triangle
(perpendicular to the base)

32. ALTITUDES

33. Concurrent at the ORTHOCENTER
ALTITUDES
Concurrent at the ORTHOCENTER .
orthocenter

34. Can be outside of a triangle.
ALTITUDES
Can be outside of a triangle.
altitude

35. Quick Review

36. MEDIAN

37. Concurrent at CENTROID (center of gravity)
MEDIAN
Concurrent at CENTROID
(center of gravity) .
Centroid

38. Angle Bisector

39. Concurrent at the INCENTER (center of circle INSCRIBED in triangle)
Angle Bisector
Concurrent at the INCENTER
(center of circle INSCRIBED in triangle)

40. Concurrent at the INCENTER (center of circle INSCRIBED in triangle)
Angle Bisector
Concurrent at the INCENTER
(center of circle INSCRIBED in triangle) .
Incenter

41. ALTITUDE

42. Concurrent at the ORTHOCENTER
ALTITUDES
Concurrent at the ORTHOCENTER .
orthocenter

43. PERPENDICULAR BISECTORS

44. Perpendicular Bisectors:
Concurrent at the Circumcenter .
circumcenter

45. Perpendicular Bisectors
Radii of circle go from circumcenter to each vertex of the triangle r r r

46. BISECTORS

47. Bisectors
Both types of bisectors (Angle Bisectors and Perpendicular Bisectors) will lead to circles.

48. Problem: Three cities want to build a park that is the same distance from each of their city centers. What should they do?
MLT
Kenmore
Shoreline

49. Which “triangle center” will be the same distance from each city center?
MLT
Kenmore
Shoreline
Shoreline

50. The CIRCUMCENTER
MLT
Kenmore
Shoreline
Shoreline

51. Which triangle segments or lines are used to find the circumcenter?
MLT
Kenmore
Shoreline
Shoreline

52. PERPENDICULAR BISECTORS
MLT
Kenmore
Shoreline
Shoreline

53. PERPENDICULAR BISECTORS (green lines) are concurrent at the CIRCUMCENTER.
MLT
Shoreline
Circumcenter
Kenmore

54. The Circumcenter is equidistant from each city center.
MLT
Circumcenter
Shoreline
Kenmore

55. The distance is the RADIUS of the circle centered at the CIRCUMCENTER.
MLT r C r r
Shoreline
Kenmore

56. Problem: Three cities want to build a toxic waste dump that is the same distance from each of their city centers. What should they do?
MLT
Kenmore
Shoreline

57. Which “triangle center” will be the same distance from each city center?
MLT
Kenmore
Shoreline

58. Which “triangle center” will be the same distance from each city center? The CIRCUMCENTER
MLT
Kenmore
Shoreline

59. Which triangle segments or lines are used to find the circumcenter?
MLT
Kenmore
Shoreline

60. Which triangle segments or lines are used to find the circumcenter
Which triangle segments or lines are used to find the circumcenter? PERPENDICULAR BISECTORS
MLT
Kenmore
Shoreline

61. PERPENDICULAR BISECTORS are concurrent at the CIRCUMCENTER.
MLT
Kenmore
Shoreline
Circumcenter

62. The Circumcenter is equidistant from each city center.
MLT
Shoreline
Kenmore
Circumcenter

63. The distance is the RADIUS of the circle centered at the CIRCUMCENTER.
MLT
Shoreline
Kenmore
radius
Circumcenter

64. The circumcenter can be outside of the triangle.
MLT
Shoreline
Kenmore
radius
Circumcenter

65. The centroid and incenter must be inside of the triangle.