Homework 2 Perpendicular And Angle Bisectors Powerpoint

Presentation on theme: "Perpendicular and Angle Bisectors"— Presentation transcript:

1 Perpendicular and Angle Bisectors
GeometryPerpendicular and Angle BisectorsCONFIDENTIAL

2 Write the equation of each line in slope-intercept from.
Warm upWrite 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 5CONFIDENTIAL

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 HYPOTHESIS CONCLUSION
TheoremsDistance and Perpendicular BisectorsTHEOREMHYPOTHESISCONCLUSIONLX1.1) Perpendicular Bisector TheoremIf a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoint of the segment.AYBNext page ->CONFIDENTIAL

5 HYPOTHESIS CONCLUSION
THEOREMHYPOTHESISCONCLUSIONL1.2) Converse of the Perpendicular Bisector TheoremIf a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.XAYBCONFIDENTIAL

6 Perpendicular Bisector Theorem
XAYBCONFIDENTIAL

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.CONFIDENTIAL

8 Applying the Perpendicular Bisector Theorem and Its Converse
Find each measure.LA) YWW7.3YW = XWYW = 7.3XZYNext page ->CONFIDENTIAL

9 B) BCB36DLA1636Next page ->CCONFIDENTIAL

10 C) PR PR = RQ 2n + 9 = 7n – 18 9 = 5n – 18 27 = 5n 5.4 = n
So PR = 2(5.4) + 9 = 19.8LSQP2n + 97n - 18RCONFIDENTIAL

11 Now you try! 1) Find each measure. L G
Given that line l is the perpendicularbisector of DE and EG = 14.6, find DG.b) Given that DE = 20.8, DG = 36.4, andEG = 36.4, find EF.DFECONFIDENTIAL

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 HYPOTHESIS CONCLUSION
TheoremsDistance and Angle BisectorsTHEOREMHYPOTHESISCONCLUSION1.3) Angle BisectorTheoremIf a point is on the bisector of an angle, then it is equidistant from the sides of the angle.ACAC = BCPB/APC ≅ /BPCNext page ->CONFIDENTIAL

14 HYPOTHESIS CONCLUSION
THEOREMHYPOTHESISCONCLUSION1.4) Converse of the Angle Bisector TheoremIf 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.AC/APC ≅ /BPCPBAC = BCCONFIDENTIAL

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.JA) LM12.8MKLM = JM / Bisector Thm.LM = Substitute 12.8 for JM.LNext page ->CONFIDENTIAL

17 B) m/ABD, given that m/ABC = 112˚
74A74BCDef. of / bisectorSubstitute 112˚ for m/ABC.Next page ->CONFIDENTIAL

18 UC) m/TSUR(6z + 14˚)ST(5z + 23˚)CONFIDENTIAL

19 Now you try!2) Find each measure.WZYXCONFIDENTIAL

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?PSRQCONFIDENTIAL

21 S is equidistant from each pair of suspension
Now you try!S is equidistant from each pair of suspensionlines. What can you conclude aboutPSRQCONFIDENTIAL

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).yA(1,5)B4x24Next page ->CONFIDENTIAL

23 yA(1,5)B4x24Next page ->CONFIDENTIAL

24 yA(1,5)B4x24Next page ->CONFIDENTIAL

25 yA(1,5)B4x24CONFIDENTIAL

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

27 Now some problems for you to practice !
CONFIDENTIAL

28 Use the diagram for Exercise.
AssessmentUse the diagram for Exercise.mSQPT1)2)CONFIDENTIAL

29 Use the diagram for Exercise.
3)ADCB4)CONFIDENTIAL

30 5) For a king post truss to be constructed
correctly, P must lie on the bisector of /JLK. How canbraces PK and PM be used to ensure that P is in theproper location?LMKJNPCONFIDENTIAL

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

32 Perpendicular and Angle Bisectors
Let’s reviewPerpendicular and Angle BisectorsWhen 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
TheoremsDistance and Perpendicular BisectorsTHEOREMHYPOTHESISCONCLUSIONLX1.1) Perpendicular Bisector TheoremIf a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoint of the segment.AYBNext page ->CONFIDENTIAL

34 HYPOTHESIS CONCLUSION
THEOREMHYPOTHESISCONCLUSIONL1.2) Converse of the Perpendicular Bisector TheoremIf a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.XAYBCONFIDENTIAL

35 Perpendicular Bisector Theorem
XAYBCONFIDENTIAL

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.LA) YWW7.3YW = XWYW = 7.3XZYNext page ->CONFIDENTIAL

38 B) BCB36DLA1636Next page ->CCONFIDENTIAL

39 C) PR PR = RQ 2n + 9 = 7n – 18 9 = 5n – 18 27 = 5n 5.4 = n
So PR = 2(5.4) + 9 = 19.8LSQP2n + 97n - 18RCONFIDENTIAL

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
TheoremsDistance and Angle BisectorsTHEOREMHYPOTHESISCONCLUSION1.3) Angle BisectorTheoremIf a point is on the bisector of an angle, then it is equidistant from the sides of the angle.ACAC = BCPB/APC ≅ /BPCNext page ->CONFIDENTIAL

42 HYPOTHESIS CONCLUSION
THEOREMHYPOTHESISCONCLUSION1.4) Converse of the Angle Bisector TheoremIf 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.AC/APC ≅ /BPCPBAC = BCCONFIDENTIAL

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.JA) LM12.8MKLM = JM / Bisector Thm.LM = Substitute 12.8 for JM.LNext page ->CONFIDENTIAL

45 B) m/ABD, given that m/ABC = 112˚
74A74BCDef. of / bisectorSubstitute 112˚ for m/ABC.Next page ->CONFIDENTIAL

46 UC) m/TSUR(6z + 14˚)ST(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?PRSQCONFIDENTIAL

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).yA(1,5)B4x24Next page ->CONFIDENTIAL

49 yA(1,5)B4x24Next page ->CONFIDENTIAL

50 yA(1,5)B4x24Next page ->CONFIDENTIAL

51 yA(1,5)B4x24CONFIDENTIAL

52 You did a great job today!
CONFIDENTIAL

Presentation on theme: "5-2 Bisectors of a Triangle"— Presentation transcript:

1 5-2 Bisectors of a Triangle
Rigor: apply the perpendicular bisector theorem, the angle bisector theorem, and their conversesRelevance: City planning and interior design

2 Exploring Perpendicular Bisectors in a triangle
Draw a triangle (any triangle) on a piece of tracing paperFold each side in half. The creases are the perpendicular bisectors of each side.What do you notice?Concurrent – 3 or more lines intersect at one point: the point of concurrency.

3 Circumcenter – the point of concurrency of the ┴ bisectors
Concurrency of Perpendicular Bisectors Theorem – the circumcenter of a ∆ isequidistant from the verticesAP = BP = CPCircumcenter can be inside, outside, or on ∆

4 Circumscribed Circles
Circumscribed circle – a circle that has all 3 vertices of a triangle on the circle with the center of circle as the circumcenter of the triangleThe prefix circum- means “around”, so a circumscribed circle goes around the triangleTurn in your core book to page 199 EX 1 and construct a circumscribed circle

5 EX 1: What are the coordinates of the circumcenter of ∆ with vertices A(2,7), B(10,7) & C(10,3)
Step 1: Graph ∆Step 2: Calculate midpoints (count if vertical or horizontal sides)Step 3: Use right angle to draw ┴ bisector

6 EX 2: City Planning

7 Angle Bisectors and Incenters
Inscribed circle – a circle that touches every side of the triangle with the incenter as its centerTurn to pg 200 in the core book and construct an inscribed circle

8 Concurrency of Angle Bisectors Theorem
Incenter – the point of concurrency of the angle bisectors; always inside the triangleTheorem: The incenter of a triangle is equidistant from the sides to the triangle.

9 EX 3: Calculate the length of the segment.
A) GE = 2x – 7;GF = x + 4What is GD?B) QN = 5x + 36;QM = 2x + 51What is QO?

10 EX 4: CampingLogan plans to go camping in a state park. The park is bordered by 3 highways, and Logan wants to pitch his tent as far away from the highways as possible. Should he set up camp at the circumcenter or the incenter of the park? Why?

11 5-2 Classwork Core book pgs 201 – 202 #1 – 3, 7 – 9
Textbook pg 323 – #9, 10, 14 – 16, 20, 22, 23, 26 – 325-2 HomeworkCore book pg 203 – 204 ALLDue Thursday for periods 1, 3, 5Due Friday for periods 2, 4, 7

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