Transcript Chap. 4-7 Medians, Altitudes and Perpendicular Bisectors

4-7 Median, Altitude, and
Perpendicular bisectors
Medians, Altitudes and Perpendicular Bisectors
A median of a triangle is a segment from a vertex to the
midpoint of the opposite side.
A
B
C
D
Medians, Altitudes and Perpendicular
Bisectors
Every triangle has 3 medians. The 3 medians of
below.
A
B
C
D
ABC are shown
Medians, Altitudes and Perpendicular Bisectors
An altitude of a triangle is the perpendicular segment from a vertex to
the line that contains the opposite side
A
B
C
Medians, Altitudes and Perpendicular Bisectors
In an acute triangle, the three altitudes are
all inside the triangle.
A
B
C
D
Medians, Altitudes and Perpendicular
Bisectors
In a right triangle, two of the altitudes are parts of the triangle. They are
the legs of the right triangle. The third altitude is inside the triangle.
B
D
A
C
Medians, Altitudes and Perpendicular Bisectors
In an obtuse triangle, two of the altitudes are outside the triangle.
ABC has two altitudes outside the triangle.
A
B
C
Medians, Altitudes and Perpendicular
Bisectors
A perpendicular bisector of a segment is a line that is perpendicular to
the segment at its midpoint. In the figure below, line m is a
perpendicular bisector of AB.
m
A
B
Perpendicular Bisector Theorem
Theorem 4-5
If a point lies on the perpendicular bisector of a segment, then the point
is equidistant from the endpoints of the segment.
C
A
B
D
If CD is the perpendicular bisector of AB than AC = BC
8
Perpendicular Bisector Converse Theorem
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
C
A
B
10
Theorem 4-7
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
Z
P
B
Y
C
Theorem 4-8
Theorem 4-8
If a point is equidistance from the side of an angle, then the point lies on
the bisector of an angle
A
X
Z
P
B
Y
C