Medians, Altitudes and Perpendicular Bisectors

1
Section 4-7

• Medians, Altitudes and Perpendicular Bisectors

2
Median

• connects the vertex to the midpoint of the
  opposite side

3
Thus, every triangle has three medians.
4
Altitude

• the perpendicular segment from a vertex to a line
  that contains the opposite side

B
F
D
C
A
E
5

• For obtuse triangles, two altitudes fall outside
  the figure.
• You extend the base of the triangle so that it
  intersects the altitude at a right angle.

B
B
C
A
C
H
A
J
6
For obtuse triangles, there is still one altitude
in the triangle.
B
K
C
A
7
In Right Triangles Two of the altitudes lie on
the legs of the triangle. The 3rd is inside.
8
Perpendicular bisector of a
segment

• is a line (or ray or segment) that is
  perpendicular to the bisector at its midpoint

9
In a given plane, there is exactly one
perpendicular to a segment at its midpoint
M
l
K
J
10
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

11
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
X
12
Theorem 4-7

• If a point lies on the bisector of an angle,
  then the point is equidistant from the sides of
  the angle.

13
Theorem 4-8
If a point is equidistant from the sides of an
angle, then the point lies on the bisector of the
angle.
14
Always, Sometimes or Never?
Always

• An altitude is _______ perpendicular to the
  opposite side?
• A median is _______ perpendicular to the
  opposite side?
• An altitude is ________an angle bisector?

Sometimes
Sometimes
15
Always, Sometimes or Never?
Sometimes

• An angle bisector is ___________ perpendicular
  to the opposite side.
• A perpendicular bisector is ________
  perpendicular to a segment at its midpoint.
• A perpendicular bisector of a segment is
  __________ equidistant from the endpoints of the
  segment.

Always
Always