Perpendicular and Angle Bisectors Geometry Perpendicular and Angle Bisectors CONFIDENTIAL

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 CONFIDENTIAL

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

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

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 CONFIDENTIAL

Perpendicular Bisector Theorem X A Y B CONFIDENTIAL

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

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

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

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

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

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

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

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 CONFIDENTIAL

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

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

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

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

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

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 CONFIDENTIAL

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 CONFIDENTIAL

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 Next page -> CONFIDENTIAL

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

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

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

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

Now some problems for you to practice ! CONFIDENTIAL

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

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

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

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

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

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

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 CONFIDENTIAL

Perpendicular Bisector Theorem X A Y B CONFIDENTIAL

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

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

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

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

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

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

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 CONFIDENTIAL

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

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

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

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

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

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 Next page -> CONFIDENTIAL

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

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

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

You did a great job today! CONFIDENTIAL