14-1 to 14-3: Special Segments

14-1 to 14-3: 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
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Assignment:
P. 306-309: 2, 3, 5, 9, 10,
11, 13, 21, 26, 34, 35
P. 313-316: 2, 3-13 odd,
12, 31, 34, 39-41
P. 322-325: 2, 13, 15, 17,
19, 21, 36, 38, 46, 47
Challenge Problems
Bring Functional
Compass
Objective 2
You will be
able to discover,
use, and prove
various theorems
about perpendicular
bisectors and angle bisectors
Paper Folding Activity 1
1. On a piece of
patty paper, use a
ruler to draw AB.
Paper Folding Activity
2. Now fold the piece
of paper so that
point A lies
coincides with (lies
directly on top of)
point B.
Paper Folding Activity
3. Unfold the paper
and label point M
where the crease
intersects the
segment. This
crease is called the
perpendicular
bisector.
Perpendicular Bisector
A segment, ray,
line, or plane that is
perpendicular to a
segment at its
midpoint is called a
perpendicular
bisector.
Paper Folding Activity
4. Put a point P
somewhere on the
perpendicular
bisector. Now
compare the
lengths PA and
PB.
P
Equidistant
A point is equidistant
from two figures if the
point is the same
distance from each
figure.
Examples: midpoints
and parallel lines
Investigation 1
In this GSP
demonstration, we
will discover two
important properties
of perpendicular
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.
Example 2
BD is the perpendicular bisector of AC. Find
AD.
Example 3
Find the values of x and y.
Angle Bisector
An angle bisector is
a ray that divides an
angle into two
congruent angles.
Paper Folding Activity 2
Step 1: On a patty
paper, draw a
large acute angle.
Label it PQR.
Paper Folding Activity 2
Step 2: Fold your patty
paper over so that
QP and QR coincide.
Crease your patty
paper along the fold.
Notice that you are
not necessarily trying
to put point P on R.
You’re just lining up
the rays.
Paper Folding Activity 2
Step 3: Unfold your
patty paper. Draw
a ray with endpoint
Q along the
crease. Is the ray
the angle bisector
of <PQR?
Paper Folding Activity 2
Step 4: Place a point on your angle bisector.
Label it A. Compare the perpendicular
distances to the two sides.
Paper Folding Activity 1
Step 5: Compare this
distance with the
distance to the other
side by repeating the
process on the other
ray.
What do you notice about the
two distances from a point
on the angle bisector to the
sides of the angle?
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?
Example 5
Find the value of x.
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?
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?
14-1 to 14-3: 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
•
•
•
•
•
Assignment:
P. 306-309: 2, 3, 5, 9, 10,
11, 13, 21, 26, 34, 35
P. 313-316: 2, 3-13 odd,
12, 31, 34, 39-41
P. 322-325: 2, 13, 15, 17,
19, 21, 36, 38, 46, 47
Challenge Problems
Bring Functional
Compass