1. Perpendicular and Angle Bisectors
Geometry
Perpendicular and Angle Bisectors
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2. Write the equation of each line in slope-intercept from.
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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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.
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4. HYPOTHESIS CONCLUSION
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
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5. HYPOTHESIS CONCLUSION
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
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6. Perpendicular Bisector TheoremX A Y B
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7. A locus is a set of point that satisfies a given condition
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.
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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
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9. B) BC B 36 D L A 16 36
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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
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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
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12. Remember that the distance between a point and a line is the length of the perpendicular segment from the point to the line.
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13. HYPOTHESIS CONCLUSION
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
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14. HYPOTHESIS CONCLUSION
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
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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.
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16. Applying the Angle Bisector Theorems
Find each measure. J
A) LM
12.8 M K
LM = JM / Bisector Thm.
LM = Substitute 12.8 for JM. L
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17. B) m/ABD, given that m/ABC = 112˚74 A 74 B C
Def. of / bisector
Substitute 112˚ for m/ABC.
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18. U
C) m/TSU R
(6z + 14˚) S T
(5z + 23˚)
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19. Now you try!
2) Find each measure. W Z Y X
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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 S R Q
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21. S is equidistant from each pair of suspension
Now you try!
S is equidistant from each pair of suspension
lines. What can you conclude about P S R Q
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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 2 4
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23. y A
(1,5) B 4 x 2 4
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24. y A
(1,5) B 4 x 2 4
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25. y A
(1,5) B 4 x 2 4
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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).
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27. Now some problems for you to practice !
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28. Use the diagram for Exercise.
Assessment
Use the diagram for Exercise. m S Q P T 1) 2)
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29. Use the diagram for Exercise.3) A D C B 4)
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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
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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) ) U(2,-6), V(4,0)
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32. Perpendicular and Angle Bisectors
Let’s review
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

33. HYPOTHESIS CONCLUSION
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. HYPOTHESIS CONCLUSION
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
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35. Perpendicular Bisector TheoremX A Y B
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36. A locus is a set of point that satisfies a given condition
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
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38. B) BC B 36 D L A 16 36
Next page -> C
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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
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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. HYPOTHESIS CONCLUSION
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. HYPOTHESIS CONCLUSION
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
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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 = Substitute 12.8 for JM. L
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45. B) m/ABD, given that m/ABC = 112˚74 A 74 B C
Def. of / bisector
Substitute 112˚ for m/ABC.
Next page ->
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46. U
C) m/TSU R
(6z + 14˚) S T
(5z + 23˚)
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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 2 4
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49. y A
(1,5) B 4 x 2 4
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50. y A
(1,5) B 4 x 2 4
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51. y A
(1,5) B 4 x 2 4
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52. You did a great job today!
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