Transcript perpendicular bisector

Equidistant
A point is equidistant
from two figures if
the point is the
same distance from
each figure.
Examples: midpoints
and parallel lines
5.2-5.4: Special Segments
Objectives:
1. To use and define perpendicular
bisectors, angle bisectors, medians, and
altitudes
2. To discover, use, and prove various
theorems about perpendicular bisectors
and angle bisectors
Perpendicular Bisector Theorem
In a plane, if a point is
on the perpendicular
bisector of a
segment, then it is
equidistant from the
endpoints of the
segment.
Converse of Perpendicular Bisector Theorem
In a plane, if a point is
equidistant from the
endpoints of a
segment, then it is
on the perpendicular
bisector of the
segment.
Example 1
Plan a proof for the Perpendicular Bisector
Theorem.
1. AB CP
Given
~
2. AP = PB
Def. of Bisector
3. <CPB &
Def. of Perp.
<CPA R rt. <s
~
4. <CPB=<CPA
All rt. <s R ~=
5. CP ~
= CP
Reflexive
~
6. CAP = CBP SAS
7. CA ~= CB
CPCTC
Example 2
BD is the perpendicular bisector of AC. Find
AD.
X=7
AD=35
Example 3
Find the values of x and y.
x=12
y=8
Angle Bisector
An angle bisector is a
ray that divides an
angle into two
congruent angles.
Angle Bisector Theorem
If a point is on the bisector of an angle, then
it is equidistant from the two sides of the
angle.
Example 4
A soccer goalie’s position relative to the ball
and goalposts forms congruent angles, as
shown. Will the goalie have to move
farther to block a shot toward the right goal
post or the left one? Answer in your notebook
Example 5
Find the value of x.
x=5
Example 6
Find the measure of <GFJ.
It’s not the Angle Bisector Theorem that could help
us answer this question. It’s the converse. If it’s true.
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 the angle.
Example 7
For what value of x does P lie on the bisector
of <A?
x=4
Special Triangle Segments
B
A
Perpendicular Bisector
Both perpendicular bisectors and angle
bisectors are often associated with
triangles, as shown below. Triangles have
two other special segments.
B
C
A
C
Median
Median
A median of a triangle
is a segment from a
vertex to the midpoint
of the opposite side of
the triangle.
Altitude
Altitude
An altitude of a
triangle is a
perpendicular
segment from a
vertex to the
opposite side or to
the line that
contains that side.
The length of the altitude is the height of the triangle.
Example 8
Is it possible for any of the
aforementioned special
segments to be
identical?
In other words, is there a
triangle for which a
median, an angle
bisector, and an altitude
are all the same?