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Geometry Perpendicular and Angle Bisectors

Geometry Perpendicular and Angle Bisectors. Warm up. Write the equation of each line in slope-intercept from. 1) The line through the points (1,-1) and (2, -9) 2) The line with slope -0.5 through (10, -15) 3) The line with x-intercept -4 and y-intercept 5.

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Geometry Perpendicular and Angle Bisectors

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  1. Geometry Perpendicular and Angle Bisectors CONFIDENTIAL

  2. Warm up Write the equation of each line in slope-intercept from. 1) The line through the points (1,-1) and (2, -9) 2) The line with slope -0.5 through (10, -15) 3) The line with x-intercept -4 and y-intercept 5 CONFIDENTIAL

  3. Perpendicular and Angle Bisectors When a point is the same distance from two or more objects, the point is said to be equidistant from the objects. Triangle congruence theorems can be used to prove theorems about equidistant points. CONFIDENTIAL

  4. Theorems Distance and Perpendicular Bisectors THEOREM HYPOTHESIS CONCLUSION L X 1.1) Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoint of the segment. A Y B Next page -> CONFIDENTIAL

  5. THEOREM HYPOTHESIS CONCLUSION L 1.2) Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. X A Y B CONFIDENTIAL

  6. Perpendicular Bisector Theorem L X A Y B CONFIDENTIAL

  7. A locus is a set of point that satisfies a given condition. The perpendicular bisector of a segment can be defined as the locus of points in a plane that are equidistant from the endpoints of the segment. CONFIDENTIAL

  8. Applying the Perpendicular Bisector Theorem and Its Converse Find each measure. L A) YW W 7.3 YW = XW YW = 7.3 X Z Y Next page -> CONFIDENTIAL

  9. B) BC B 36 D L A 16 36 Next page -> C CONFIDENTIAL

  10. C) PR PR = RQ 2n + 9 = 7n – 18 9 = 5n – 18 27 = 5n 5.4 = n So PR = 2(5.4) + 9 = 19.8 L S Q P 2n + 9 7n - 18 R CONFIDENTIAL

  11. Now you try! 1) Find each measure. L G • Given that line l is the perpendicular • bisector of DE and EG = 14.6, find DG. • b) Given that DE = 20.8, DG = 36.4, and • EG = 36.4, find EF. D F E CONFIDENTIAL

  12. Remember that the distance between a point and a line is the length of the perpendicular segment from the point to the line. CONFIDENTIAL

  13. Theorems Distance and Angle Bisectors THEOREM HYPOTHESIS CONCLUSION 1.3) Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. A C AC = BC P B /APC ≅ /BPC Next page -> CONFIDENTIAL

  14. THEOREM HYPOTHESIS CONCLUSION 1.4) Converse of the Angle Bisector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle. A C /APC ≅ /BPC P B AC = BC CONFIDENTIAL

  15. Based on those theorems, an angle bisector can be defined as the locus of all points in the interior of the angle that are equidistant from the sides of the angle. CONFIDENTIAL

  16. Applying the Angle Bisector Theorems Find each measure. J A) LM 12.8 M K LM = JM / Bisector Thm. LM = 12.8 Substitute 12.8 for JM. L Next page -> CONFIDENTIAL

  17. B) m/ABD, given that m/ABC = 112˚ D 74 A 74 B C Def. of / bisector Substitute 112˚ for m/ABC. Next page -> CONFIDENTIAL

  18. U C) m/TSU R (6z + 14˚) S T (5z + 23˚) CONFIDENTIAL

  19. Now you try! 2) Find each measure. W Z Y X CONFIDENTIAL

  20. Parachute Application Each pair of suspension lines on a parachute are the same length and are equally spaced from the center of the chute. How do these lines keep the sky diver centered under the parachute? P R S Q CONFIDENTIAL

  21. Now you try! • S is equidistant from each pair of suspension • lines. What can you conclude about P R S Q CONFIDENTIAL

  22. Writing Equations of Bisectors in Coordinate Plane Write an equation in point – slope form for the perpendicular bisector of the segment with endpoints A(-1,6) and B(3,4). y A (1,5) B 4 x 0 2 4 Next page -> CONFIDENTIAL

  23. y A (1,5) B 4 x 0 2 4 Next page -> CONFIDENTIAL

  24. y A (1,5) B 4 x 0 2 4 Next page -> CONFIDENTIAL

  25. y A (1,5) B 4 x 0 2 4 CONFIDENTIAL

  26. Now you try! 4) Write an equation in point-slope from for the perpendicular bisector of the segment with endpoints P(5,2) and Q(1,-4). CONFIDENTIAL

  27. Now some problems for you to practice ! CONFIDENTIAL

  28. Assessment Use the diagram for Exercise. 1) m 2) T Q P S CONFIDENTIAL

  29. Use the diagram for Exercise. A 3) D 4) B C CONFIDENTIAL

  30. 5) For a king post truss to be constructed correctly, P must lie on the bisector of /JLK. How can braces PK and PM be used to ensure that P is in the proper location? L M K J N P CONFIDENTIAL

  31. Write an equation in point-slope from for the perpendicular bisector of the segment with the given endpoints. 6) M(-5,4), N(1,-2) 7) U(2,-6), V(4,0) CONFIDENTIAL

  32. Let’s review Perpendicular and Angle Bisectors When a point is the same distance from two or more objects, the point is said to beequidistant from the objects. Triangle congruence theorems can be used to prove theorems about equidistant points. CONFIDENTIAL

  33. Theorems Distance and Perpendicular Bisectors THEOREM HYPOTHESIS CONCLUSION L X 1.1) Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoint of the segment. A Y B Next page -> CONFIDENTIAL

  34. THEOREM HYPOTHESIS CONCLUSION L 1.2) Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. X A Y B CONFIDENTIAL

  35. Perpendicular Bisector Theorem L X A Y B CONFIDENTIAL

  36. A locus is a set of point that satisfies a given condition. The perpendicular bisector of a segment can be defined as the locus of points in a plane that are equidistant from the endpoints of the segment. CONFIDENTIAL

  37. Applying the Perpendicular Bisector Theorem and Its Converse Find each measure. L A) YW W 7.3 YW = XW YW = 7.3 X Z Y Next page -> CONFIDENTIAL

  38. B) BC B 36 D L A 16 36 Next page -> C CONFIDENTIAL

  39. C) PR PR = RQ 2n + 9 = 7n – 18 9 = 5n – 18 27 = 5n 5.4 = n So PR = 2(5.4) + 9 = 19.8 L S Q P 2n + 9 7n - 18 R CONFIDENTIAL

  40. Remember that the distance between a point and a line is the length of the perpendicular segment from the point to the line. CONFIDENTIAL

  41. Theorems Distance and Angle Bisectors THEOREM HYPOTHESIS CONCLUSION 1.3) Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. A C AC = BC P B /APC ≅ /BPC Next page -> CONFIDENTIAL

  42. THEOREM HYPOTHESIS CONCLUSION 1.4) Converse of the Angle Bisector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle. A C /APC ≅ /BPC P B AC = BC CONFIDENTIAL

  43. Based on those theorems, an angle bisector can be defined as the locus of all points in the interior of the angle that are equidistant from the sides of the angle. CONFIDENTIAL

  44. Applying the Angle Bisector Theorems Find each measure. J A) LM 12.8 M K LM = JM / Bisector Thm. LM = 12.8 Substitute 12.8 for JM. L Next page -> CONFIDENTIAL

  45. B) m/ABD, given that m/ABC = 112˚ D 74 A 74 B C Def. of / bisector Substitute 112˚ for m/ABC. Next page -> CONFIDENTIAL

  46. U C) m/TSU R (6z + 14˚) S T (5z + 23˚) CONFIDENTIAL

  47. Parachute Application Each pair of suspension lines on a parachute are the same length and are equally spaced from the center of the chute. How do these lines keep the sky diver centered under the parachute? P R S Q CONFIDENTIAL

  48. Writing Equations of Bisectors in Coordinate Plane Write an equation in point – slope form for the perpendicular bisector of the segment with endpoints A(-1,6) and B(3,4). y A (1,5) B 4 x 0 2 4 Next page -> CONFIDENTIAL

  49. y A (1,5) B 4 x 0 2 4 Next page -> CONFIDENTIAL

  50. y A (1,5) B 4 x 0 2 4 Next page -> CONFIDENTIAL

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