Download Do Now: angle = D Right A B C D E <AEC <BEC <AED <DEB x y L

CC Geometry 10R
Aim 9: How do we construct a perpendicular bisector?
18°
Do Now:
62
1. Complete: An angle bisector is a ray (line/segment)
angle
that divides an ________
into
= or ≅ parts.
two ___
2. Construct and label BD, the bi sector of ≮ABC.
D
A
B
C
Relevant Vocabulary
Two lines (segments, rays) are PERPENDICULAR (
Right
_________________
angle.
Ex. a) Draw and label: AB
) if they intersect to form a
CD , AB and CD intersect at E.
C
b) Name all right angles in the diagram you drew.
<AEC
<BEC
<AED
E
<DEB
A
B
D
congruent
The MIDPOINT of a segment is a point that divides a segment into 2 = or _____ parts.
Ex. Draw and label: L is the midpoint of XY.
x
y
L
midpoint
A SEGMENT BISECTOR passes through the ____________
of a segment.
A
Ex. Draw and label: AB is a segment bisector of XY,
but AB is not
to XY.
X
Y
B
The PERPENDICULAR BISECTORof a line segment is perpendicular to a segment
at its midpoint.
Ex. Draw and label: AB is the perpendicular bisector XY;
M is the midpoint of XY.
A
X
M
B
Y
CONSTRUCTION #4: CONSTRUCT THE PERPENDICULAR BISECTOR
OF A LINE SEGMENT
Experiment with your construction tools to establish a construction that results
in the perpendicular bisector. [Hint: Use what you know about constructing an
http://www.mathsisfun.com/geometry/construct­linebisect.html
equilateral triangle.]
Steps
1. Draw circle(arc) A, center A,
radius more than ½AB.
2. Draw circle(arc) B, center B,
using same radius as in step (1).
A
3. Label the points of intersection
created by steps (1) and (2) as
C and D.
B
4. Draw CD.
Practice:
Construct the perpendicular bisector of CD.
• Label the midpoint M and labelthe perpendicular bisector as EF.
<EMC
• Name one right angle:__________.
C
M
D
A point is EQUIDISTANT from two given points if it is the
same distance from both given points.
C
Ex a) Draw point C equidistant from points A and B.
b) Draw point D equidistant from points A and B.
c) Draw point E equidistant from points A and B.
d) How many points could you draw equidistant
from A and B?
A
D
B
E
Activity: CDE is the perpendicular bisector of AB. (C, D, and E are collinear.)
Using your compass, what conclusion can you make about the following
pairs of segments?
Each point is Equidistant 1) AC and BC
from points A and B
2) AD and BD
3) AE and BE
Based on your findings, fill in the observation below.
Any point on the perpendicular bisector of a line
Equidistant
segment is _____________________
from the
endpoints of the line segment.
CONSTRUCTION #5: CONSTRUCT A PERPENDICULAR TO A LINE
FROM A POINT NOT ON THE LINE
line to line l from a point X not on line l .
We will now construct a perpendicular
Complete the construction and the steps of the construction outlined below.
X
l C
B
D
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Draw circle(arc) X so that the circle intersects line l in two points.
Label the two points of intersection as B and C.
Draw circle(arc) B and circle(arc) C, same radius as Step 1.
Label the intersection of circle B and circle C as D .
XD
Draw the perpendicular bisector: line _______.
Exercises
1. Divide segment AB into 4 segments of equal length.
B
A
2. Construct parallel lines l1 and
Step 1: Construct line
Step 2: Construct line
l2
as follows:
l3 which will be perpendicular to line l1 from point A
l2 which will be perpendicular to l3 through point A.
(Hint: This is the same as bisecting a straight angle.)
A
l1
3. Here is another method for constructing a line parallel to a given line through
a point not on the line, not using perpendicular lines.
Using the construction for copying an angle, construct a line parallel to line L
through point P.
P
L
4a) Construct the perpendicular bisector of BC.
b) Construct the angle bisector of ≮B.
C
A
B
Let's Sum it Up!!
A perpendicular bisector of a segment passes
midpoint
through the ______________of
the segment
Right angles with the segment.
and forms ________
(Mark the diagram to show this.)
A
A point A is said to be equidistant from two
different points B and C if AB = AC.
(Mark the diagram to show this.)
B
C
Construction of a perpendicular to a line from a point not on the line -
arcs (not full circles)
A
step 3
l step 1
step 2
step 2