Exploring Perpendicular Bisectors
A
B
A
Draw a segment and label the
endpoints A and B.
Y
B
Highlight the line and construct
the midpoint. Label it Y.
X
A
Y
B
Highlight the line and the midpoint
Y and construct the perpendicular
line. This line is the perpendicular
bisector of AB!
A
Y
Place a point on the perpendicular
bisector and label it X.
Find the length of AX and BX.
Record the measurements here.
X
AX= _____________
A
Y
B
B
BX= _____________
Construct AX and BX.
Explore!
1. Move X up and down the perpendicular bisector. What do you notice about AX
and BX?
________________________________________________________________
2. Make a conjecture…what do you think will always be true about a point on the
perpendicular bisector?
_______________________________________________________________
Discovery!
You have just discovered the Perpendicular Bisector Theorem!! This theorem also has
a useful converse! (FYI equidistant means the same distance)
Here is an example of how it is used!
Now you try!
Exploring Angle Bisectors
A
A
P
P
B
Construct 3 points and label them A, B and P
B
Construct a ray from P to A and from P to B so
you have an angle
A
A
C
P
P
B
Highlight points A, P and B (in that order) and
construct Angle Bisector
B
Place a point on the angle bisector and label it C.
D
Reminder: The distance between
a point and a line is the length of
the perpendicular segment from
the point to the line.
A
C
P
B
We want to find the distance from C to ray PA,
so we need to construct a perpendicular line
from C to ray PA. To do so, Highlight C and the
ray PA (not points P & A) and construct
perpendicular line. Place a point, D, at the
intersection.
D
A
C
P
B
E
Do the same for the other side of
the angle. Label that intersection E
Find the distance between C & D
and the dist. Between C & E.
Record the measurements here.
CD= _____________
CE= _____________
Explore!
1. Move C up and down the angle bisector. What do you notice about CD and CE?
________________________________________________________________
2. Make a conjecture…what do you think will always be true about a point on the
angle bisector?
_______________________________________________________________
Discovery!
You have just discovered the Angle Bisector Theorem!! This theorem also has a useful
converse! (FYI equidistant means the same distance)
Here is an example of how it is used!
Now you try!