objective: after studying this lesson you will be able to recognize the relationship between...
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Objective: After studying this lesson you will be able to
recognize the relationship between equidistance and perpendicular bisection.
4.4 The Equidistance Theorems
Definition The distance between two points is the length of the shortest path joining them.
Postulate A line segment is the shortest path between two points
If two points A and B are the same distance from a third point Z, then Z is said to be equidistant to A and B.
B A
Z
BC
A
D
B
C
A
DB
C
A
D
What do these drawings have in common?
A and B are equidistant from points C and D. We could prove that line AB is the perpendicular bisector of segment CD with the following theorems.
Definition The perpendicular bisector of a segment is the line that bisects and is perpendicular to the segment.
Theorem If 2 points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of that segment.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
Given:
Prove:
If 2 angles are both supplementary and congruent , then
they are right angles.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
If a line divides a segment into 2 congruent segments, it bisects it.
CPCTC
If 2 lines intersect to form right angles they are perpendicular.Combination of steps 9 and 12
B
C
D
EA
CDBCADAB ,
AC is the bisector of BD.�������������� �
Given
Given
SSS (1,2,3)
SAS (1,5,6)
Reflexive Property
Reflexive Property
CPCTC
CPCTC
AB ADBC CDAC ACABC ADC BAC DAC AE AEABE ADE BE DE
AC is the bisector of BD.�������������� �
AC bisects BD�������������� �
AEB AED are rt. 'sAEB AED
AC BD.�������������� �
Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of that segment.
1.
2.
3.
4.
5.
Given:
Prove:
1.
2.
3.
4.
5.
3 4 bis. AE BD
�������������� �B
C
E
A
D
1 2
13
2
4
If 2 points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of that segment.
1.
2.
3.
4.
Given:
Prove:
1.
2.
3.
4.
B CE
A
is isoscelesABC
AE BC�������������� �
AB AC is the midpoint of E BC
If 2 points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of that segment.
1.
2.
3.
4.
Given:
Prove:
1.
2.
3.
4.
BC
E
A
D
AB ADBC CDBE ED
A
If 2 points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of that segment.
A point of the perpendicular bisector of a segment is equidistant from the endpoints of the segment
Summary:
Define equidistant in your own words and summarize how we used it in proofs.
Homework: worksheet
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