Perpendicular bisectors
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Transcript Perpendicular bisectors
Do Now
Take a ruler from the bookshelf.
Take out a compass.
Draw an obtuse angle.
Construct (using only a compass and straightedge) a
duplicate angle.
PERPENDICULAR BISECTORS
Perpendicular Bisectors—Terms
A segment bisector—a line, ray, or segment that
passes through the midpoint of a segment.
Cuts
the line segment in half
Perpendicular lines—intersect at a right angle.
Perpendicular bisector—passes through the
midpoint of a segment at a right angle.
Equidistant—the same distance
Constructing Perpendicular Bisectors
Step 1: Draw a line segment. Set your compass to
more than half the distance between the two
endpoints.
Step 2: Using one endpoint as center, swing an arc
on both sides of the segment.
Step 3: Using the same compass setting, swing an
arc from the other endpoint to intersect each arc.
Step 4: Mark your two intersection points and
connect them.
Perpendicular Bisector Conjecture
If a point is on the perpendicular bisector of a
segment, then it is _________ from the endpoints.
equidistant
Converse of Perpendicular Bisector
Conjecture
If a point is equidistant from the endpoints of a
segment, then it is on the _______________of the
segment.
perpendicular bisector
Also true!
Practice
Draw and label AB. Construct the perpendicular
bisector of AB.
Practice
Draw and label QD. Construct perpendicular
bisectors to divide QD into four congruent segments.
Hint: To divide it into two congruent segments, you
need a perpendicular bisector. Can you divide each
of those segments again?
Perpendicular Postulate
If there is a line and a point not on the
line, then there is exactly one line through
the point perpendicular to the given line.
Exploring Slopes
What do you notice about the slopes of
parallel lines?
What do you notice about the slopes of
perpendicular lines?
Slopes of Parallel Lines
Find the slope of each line.
What do you notice?
Parallel Lines
have equal slopes
Equations of Parallel Lines
Are these lines
parallel?
y=3x
+8
y=3x – 4
How do you know?
Slopes of Perpendicular Lines
Find the slope of
each of these lines.
What do you notice?
Perpendicular lines
have opposite
reciprocal slopes.
(Both opposite AND
reciprocal)
Equations of Perpendicular Lines
Are these lines
perpendicular?
y= 5x + 7
y= 5x – 2
NO!
y= ½ x – 3
y= - ½ x – 9
NO!
y= ¼ x
y= 4x + 7
NO!
y= -⅓x + 2
y= 3x – 4
Derive the Expression for Slopes of
Perpendicular Lines
If this were the slope of a line, what would be the slope
of a line perpendicular to it?
3
1/6
-8
-1/2
3/4
-t
a/b
m
Slope and Midpoint
To find slope:
To find midpoint:
Practice
Line segment AB starts at A (-4, 1) and ends at B (0,
3). Line segment CD starts at C (-1, 5) and ends at D
(1, 1).
1. Determine if these lines are perpendicular
bisectors.
A.
B.
Hint: They would have to be perpendicular (opposite
reciprocal slopes)
Hint: They would also have to be bisectors (have the
same midpoint)
Before the Exit Slip
If you finish early, you may take some time to finish
the 3.1 worksheet or the 3.2 worksheet.
Save about 12 minutes for the exit slip.
Today’s Objectives
Duplicate a line segment, an angle and a polygon
Construct perpendicular bisectors and midpoints
Make conjectures about perpendicular bisectors
Use Problem Solving skills
Exit Slip
For all exercises, do not erase your construction marks.
For #1-2, Determine if these two lines are parallel,
perpendicular, or intersecting. How do you know?
1.
y=7x+3,
y=-1/7x – 6
2.
y=1/3x – 8, y=-1/3x – 3
3.
Draw a line segment. Label it PQ, then construct its
perpendicular bisector.
4. Line segment AB starts at A (1, 2) and ends at B (4, 0).
Line segment CD starts at C (.5, -2) and ends at D (4.5,
4).
A.
B.
Determine if these lines are perpendicular bisectors.
Explain your reasoning.