Perpendicular bisectors

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Do Now
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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
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A segment bisector—a line, ray, or segment that
passes through the midpoint of a segment.
 Cuts
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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
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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
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If a point is on the perpendicular bisector of a
segment, then it is _________ from the endpoints.
equidistant
Converse of Perpendicular Bisector
Conjecture
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If a point is equidistant from the endpoints of a
segment, then it is on the _______________of the
segment.
perpendicular bisector
Also true!
Practice
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Draw and label AB. Construct the perpendicular
bisector of AB.
Practice
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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
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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
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What do you notice about the slopes of
parallel lines?
What do you notice about the slopes of
perpendicular lines?
Slopes of Parallel Lines
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Find the slope of each line.
What do you notice?
Parallel Lines
have equal slopes
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Equations of Parallel Lines
Are these lines
parallel?
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 y=3x
+8
 y=3x – 4
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How do you know?
Slopes of Perpendicular Lines
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Find the slope of
each of these lines.
What do you notice?
Perpendicular lines
have opposite
reciprocal slopes.
(Both opposite AND
reciprocal)
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Equations of Perpendicular Lines
Are these lines
perpendicular?
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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
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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
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To find slope:
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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
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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
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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.

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