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Lesson 4-7:
Medians, Altitudes, and Perpendicular Bisectors
(page 152)
Essential Questions
Can you construct a proof
using congruent triangles?
MEDIAN (of a triangle): a segment from a vertex to the
of the opposite side.
midpoint
B
A
C
MEDIAN (of a triangle): a segment from a vertex to the midpoint
of the opposite side.
B
A
C
MEDIAN (of a triangle): a segment from a vertex to the midpoint
of the opposite side.
B
A
C
The name given to the point of intersection for the three medians is
a bonus question on the TEST
_____________________________________.
B
A
C
The three medians
for an obtuse triangle.
Y
X
Z
ALTITUDE (of a triangle): the perpendicular segment from a vertex
to the line containing the opposite side.
B
A
C
ALTITUDE (of a triangle): the perpendicular segment from a vertex
to the line containing the opposite side.
B
A
C
ALTITUDE (of a triangle): the perpendicular segment from a vertex
to the line containing the opposite side.
B
A
C
The name given to the point of intersection for the three altitudes is
a bonus question on the TEST
_____________________________________.
B
A
C
Y
The three altitudes
for an obtuse triangle.
X
YES …
they do intersect!
Z
E
The three altitudes
for a right triangle.
D
F
PERPENDICULAR BISECTOR (of a segment): a line, ray, or segment
that is perpendicular to the segment at its
midpoint .
B
A
C
PERPENDICULAR BISECTOR (of a segment): a line, ray, or segment
that is perpendicular to the segment at its midpoint .
B
A
C
PERPENDICULAR BISECTOR (of a segment): a line, ray, or segment
that is perpendicular to the segment at its midpoint .
B
A
C
The name given to the point of intersection for the three ⊥-Bisectors is
a bonus question on the TEST
_____________________________________.
B
A
C
The three ⊥-Bisectors
for an obtuse triangle.
Y
X
Z
BISECTOR of an ANGLE: the ray that divides the angle into
two
congruent
adjacent angles.
B
A
C
BISECTOR of an ANGLE: the ray that divides the angle into
two congruent adjacent angles.
B
A
C
BISECTOR of an ANGLE: the ray that divides the angle into
two congruent adjacent angles.
B
A
C
The name given to the point of intersection for the three ∠-Bisectors is
a bonus question on the TEST
_____________________________________.
B
A
C
The three ∠-Bisectors
for an obtuse triangle.
Y
X
Z
Theorem 4-5
If a point lies on the perpendicular bisector of a segment, then the
point is
equidistant
Given: l is the ⊥-bisector of BC
A is on l
Prove:
from the endpoints of the segment.
A
AB = AC
X
B
C
l
Given: l is the ⊥-bisector of BC
A is on l
Prove:
A
AB = AC
X
B
C
l
Proof:
To prove this theorem, the following triangles must be proven congruent …
∆ ABX ≅ ∆ ACX , by SAS
then AB
= AC by CPCTC.
Postulate,
Theorem 4-6
If a point is equidistant from the endpoints of a segment, then the
point lies on the
perpendicular
bisector of the segment.
A
B
C
A
B
Proof:
X
C
To prove this theorem, the following triangles must be proven congruent.
1st way:
Draw AX ^ BC, then …
ABX @ ACX, by HL (AAS) Theorem
and BX @ CX by CPCTC,
\ AX is the ^ -bisector of BC.
Given:
Prove:
AB = AC
A
A is on the ^ -bisector of BC
1
B
2
X
C
Theorem 4-6 is the converse of Theorem 4-5
and can be combined into a biconditional.
A point is on the perpendicular bisector of a
if and only if it is equidistant
segment ______________
from the endpoints of the segment.
DISTANCE from a POINT to a LINE (or a plane): the length of the
perpendicular
segment from the point to the line (or plane).
R
NO!
NO!
t
Theorem 4-7
If a point lies on the bisector of an angle,
then the point is equidistant from the sides of the angle.
A
X
Prove:
Z
P
PX = PY
B
Y
C
A
X
Prove:
PX = PY
Proof:
P
B
Statements
1.
Z
BZ bisects ÐABC; PX ^ BA; PY ^ BC
Y
C
Reasons
Given ________________________
____________________________________
___________
Def. of Angle Bisector
_____________________________________________
3.
∠PBX ∠PBY
∠BXP ∠BYP
____________________________________
4.
____________________________________
5.
____________________________________
6.
____________________________________
2.
____________________________________
BP @ BP
∆ BXP ∆ BYP
PX @ PY or PX = PY
Def. of ⊥-lines & ≅ ∠’s
_____________________________________________
Reflexive Property
_____________________________________________
AAS Theorem
CPCTC
_____________________________________________
_____________________________________________
Theorem 4-8
If a point is equidistant from the sides of an angle,
then the point lies on the
bisector of the angle.
A
X
P
B
Y
C
A
X
P
B
Y
C
Theorem 4-8 is the converse of Theorem 4-7
and can be combined into a biconditional.
A point is on the bisector of an angle
if and only if it is
_______________
equidistant from the sides of the angle.
Assignment
Written Exercises on pages 156 & 157
RECOMMENDED: 1 to 12 ALL numbers
REQUIRED: 8, 9, 10, 12, 13, 19, 23
Prepare for Quiz on Lessons 4-6 and 4-7
Prepare for Test on Chapter 4: Congruent Triangles
Can you construct a proof
using congruent triangles?