geometry: chapter 5 5.1: perpendicular bisectors
TRANSCRIPT
GEOMETRY: Chapter 55.1: Perpendicular Bisectors
A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector.
A point is equidistant from two figures if the point is the same distance from each figure. Points on the perpendicular bisector of a segment are equidistant from the segment’s endpoints.
Draw and explain an example.
Theorem 5.1: Perpendicular Bisector Theorem
In a plane, if a point is one the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
Images taken from: Geometry. McDougal Littell: Boston, 2007. P. 303.
If is the bisector of , then .CP A CAB CB �������������� �
Theorem 5.2: Converse of the Perpendicular Bisector Theorem
In a plane, if a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector of a segment.
Images taken from: Geometry. McDougal Littell: Boston, 2007. P. 303.
If then lies on the bisector , of .DA D D BB A
Ex. 1:
27
Images taken from: Geometry. McDougal Littell: Boston, 2007. P. 304
In the diagram, is the perpendicular bisector of .
Find .
RS PQ
PR
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Images taken from: Geometry. McDougal Littell: Boston, 2007. P. 304.
Ex. 2:In the diagram, is the perpendicular bisector of .
a. Which lengths in the diagram are equal?
b. Is on ?
JM HK
L JM
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Images taken from: Geometry. McDougal Littell: Boston, 2007. P. 304.
Ex. 2:In the diagram, is the perpendicular bisector of .
a. Which lengths in the diagram are equal?
b. Is on ?
JM HK
L JM
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Theorem 5.3: Angle Bisector TheoremIf a point is on the
bisector of an angle,
then it is equidistant
from the two sides
of the angle.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 310.
If bisects and and ,
then .
AD BAC DB AB
DB D
DC AC
C
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Theorem 5.4: Converse of the Angle Bisector Theorem
If a point is in the interior
of an angle and is
equidistant from the
sides of the angle, then it
lies on the bisector of an angle.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 310.
If and and ,
then bisects .
DDB AB DC AC
AD BA
C
C
B D
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Ex. 3: Find the measure of angle BAD.
Images taken from: Geometry. McDougal Littell: Boston, 2007. P. 311.
Images taken from: Geometry. McDougal Littell: Boston, 2007. P. 311.
Ex. 4: Three spotlights form two congruent angles. Is the actor closer to the spotted area on the right or on the left?
Ex. 5: For what value of x does P lie on the bisector of angle A?
Images taken from: Geometry. McDougal Littell: Boston, 2007. P. 311.
5.1, p. 269, #1-13 odds, 16-26 all