Journal Chapter 5

1. Perpendicular Bisector and Theorem • Angle Bisector and Theorem • Concurrency • Concurrency of Perpendicular Bisectors • Circumcenter • Concurrency of Angle Bisectors • Incenter • Median • Centroid • Concurrency of Medians • Altitude of a Triangle • Orthocenter • Concurrency of Altitudes • Midsegment and Theorem • Longer and Shorter Sides • Angle and Triangle inequality • Indirect Proofs • Hinge Theorem • Triangle Relationships Journal Chapter 5 Kirsten Erichsen 9-5

2. What is a Perpendicular Bisector? A perpendicular bisector is a line that bisects a segment and is perpendicular to the line being bisected. The perpendicular bisector passes right through the middle of the line or segment.

3. Example 1. The Perpendicular Bisector, bisects the segment or line right through the middle. The perpendicular bisector bisects right through the middle of the segment.

4. Example 2. The Perpendicular Bisector passes through the line to make up four 90° angles. The perpendicular bisector bisects right through the middle of the segment.

5. Example 3. Remember, the perpendicular bisector bisects through the middle, so both sides of the line after they have been bisected are congruent. The perpendicular bisector bisects right through the middle of the segment.

6. Perpendicular Bisector Theorem. • If a point is on the perpendicular bisector, then it is equidistant to both endpoints of the segment. • CONVERSE: If a point is equidistant to both endpoints, then it lies on the perpendicular bisector.

7. Example 1. Since point P lies on the Perpendicular Bisector, then it is has the same measure to both sides. So in conclusion, the measurement from P to J is 6 centimeters. The measurements from point P to point J are also 6 cm.

8. Example 2. Since point P lies on the Perpendicular Bisector, then it is has the same measure to both sides. So in conclusion, the measurement from G to C is 9 centimeters. The measurement from point G to point C is 9 cm.

9. Example 3. Since point P lies on the Perpendicular Bisector, then it is has the same measure to both sides. So in conclusion, the measurement from J to L is 15 centimeters. The measurement from point J to point L is 15 cm.

10. What is an Angle Bisector? An Angle Bisector is a ray or line that cuts an angle into 2 congruent angles. The angle bisector that cuts the angle, always has to be in the interior or inside of the angle.

11. Example 1. The right angle is bisected into 2 congruent angles, both measuring 45°. The ray (purple) is the angle bisector.

12. Example 2. The obtuse angle measures 140°. Since it has been bisected, both angles measure 70°. The ray (green) is the angle bisector.

13. Example 3. The acute angle measures 45°. Since it has been bisected by a ray, then each one of the angles measures 22.5°. The ray (green) bisects the acute angle.

14. Angle Bisector Theorem. • Any point that lies on the angle bisector, is equidistant to both sides of the angle. • CONVERSE: If it is equidistant to both sides of the angle, then it lies on the angle bisector.

15. Example 1. To find the length of the other length, then follow the Angle Bisector Theorem. If it lies on the bisector then it is congruent to both sides. The length from point D to point E is 15 centimeters.

16. Example 2. Both sides are congruent if the point lies in the Angle Bisector. The length from J to K is congruent to the length of J to N. The length from point J to N is 9 centimeters.

17. Example 3. Both sides are congruent if the point lies in the Angle Bisector. The length from W to V is congruent to the length of W to X. The length from point W to point X is 35 centimeters.

18. What is Concurrency? Concurrent: it is said when three or more lines intersect at one point. The point of intersection is called the point of concurrency.

19. Example 1. All 3 lines are concurrent because they intersect at the same point of concurrency. A The point of Concurrency is point A.

20. Example 2. All 4 lines are concurrent because they intersect at the same point of concurrency. J The point of Concurrency is point J.

21. Example 3. All 4 lines are concurrent because they intersect at the same point of concurrency. L The point of Concurrency is point L.

22. Concurrency of Perpendicular Bisectors. • The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of a triangle. • This is also called the circumcenter.

23. Example 1.

24. Example 2.

25. Example 3. It is 6 cm to the endpoints, because it lies on the Perpendicular Bisector

26. What is the Circumcenter? • The circumcenter is the point of concurrency of the three bisectors of a triangle. • The circumcenter of a triangle is equidistant from the vertices of the triangle.

27. Example 1. In a right triangle the circumcenter is always meet and are going to be on the lines or segments of the triangle. The Perpendicular Bisectors starts from the Midpoint and make a straight line.

28. Example 2. In an obtuse triangle the circumcenter is always outside of the triangle no matter what. The lines always start from the midpoint and make a straight line.

29. Example 3. In an acute triangle the circumcenter is always in the inside of the triangle no matter what. You always start from the midpoint of each side and make a straight line (PB).

30. Concurrency of Angle Bisectors. • The angle bisectors are also concurrent to each other. • Since a triangle has three angles, it has three angle bisectors that are concurrent. • It can also be referred as the incenter.

31. Example 1. All of the lines in blue are concurrent because they intersect at the same place. The lines in red are congruent because they have the same lengths.

32. Example 2. The lines in red are the same lengths from the point of concurrency to the sides of the triangle.

33. Example 3. The lines in red are the same lengths from the point of concurrency to the sides of the triangle.

34. What is the Incenter? • INCENTER: it is the point of concurrency of the three angle bisectors of a triangle. • The incenter of a triangle is equidistant from the sides of the triangle. • It is always going to be inside the triangle.

35. Example 1. The incenter in an acute triangle is always going to be inside.

36. Example 2. The incenter in a right triangle is always going to be inside.

37. Example 3. The incenter in an obtuse triangle is always going to be inside.

38. What is the Median? • MEDIAN: a segment whose end points are a vertex of a triangle and a midpoint. • The midpoint has to be from the opposite side of the triangle. C D

39. Example 1. I J H K The Median of this triangle is from point K to point J, but there can be 2 more.

40. Example 2. L N O M The median being shown is from point L to point M.

41. Example 3. B D A C The Median being shown is from point D to point C.

42. What is the Centroid? • CENTROID: the point of concurrency of the three medians of a triangle. • The centroid is always in the inside of the triangle. • The centroid of a triangle is located ⅔ of the distance from each vertex to the midpoint of the opposite side.

43. Example 1. To find the centroid, bisect the angle.

44. Example 2. Connect the angle bisector to the midpoint of the opposite side.

45. Example 3. The point of intersection is the centroid of the triangle.

46. Concurrency of the Medians. All of the Medians in a triangle are concurrent to each other.

47. Altitude of Triangles. • Altitude of Triangles: a perpendicular segment from a vertex to the line containing the opposite side. • Every single triangle has three altitudes. • It can be inside, outside or on the triangle.

48. Example 1. In this triangle the altitude is 4.5 centimeters.

49. Example 2. In this triangle the altitude is 3 centimeters.

50. Example 3. In this triangle the altitude is 11 centimeters.