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ATTENTION
Video Users:
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These
videos require that you have available a means of
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Video files are lengthy and may take some time to load
depending upon your connection. Please be patient.
When the video is loading for the first time, you may
experience some choppy sound and movement. Allow the
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Remember
-- use your compass and straight edge only! |
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Perpendicular
- lines (or segments) which meet to form right angles.
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Perpendicular
from
a point ON a line |
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Given:
Point P is on a given line
Task: Construct a line through P perpendicular to the given line. |
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Directions:
1. Place your compass point on P and sweep an arc of any size
that crosses the line twice (below the line). You will be creating
(at least) a semicircle. (Actually, you may draw this arc above OR
below the line.)
2. STRETCH THE COMPASS LARGER!!
3. Place the compass point where the arc crossed the line on one side
and make a small arc below the line. (The small arc could be
above the line if you prefer.) |
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4. Without changing the span on the compass, place the compass
point where the arc crossed the line on the OTHER side and make another
arc. Your two small arcs should be crossing.
5. With your straightedge, connect the intersection of the two
small arcs to point P.
This
new line is perpendicular to the given line.
Explanation
of construction:
Remember the construction
for bisect an angle? In this construction, you have bisected the
straight angle P. Since a straight angle contains 180 degrees, you
have just created two angles of 90 degrees each. Since two right
angles have been formed, a perpendicular exists.
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Perpendicular
from
a point off a line.
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Given:
Point P is off a given line
Task: Construct a line through P perpendicular to the given line. |
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Directions:
1.
Place your compass point on P and sweep an arc of any
size that crosses the line twice.
2. Place the compass point where the arc crossed the line on one
side and make an arc ON THE OPPOSITE SIDE OF THE LINE.
3. Without changing the span on the compass, place the compass
point where the arc crossed the line on the OTHER side and make another
arc. Your two new arcs should be crossing on the opposite side of
the line. |
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4. With your straightedge, connect the intersection of the two new
arcs to point P.
This
new line is perpendicular to the given line.
Explanation of
construction:
To
understand the explanation, some additional labeling will be
needed. Label the point where the arc crosses the line as points
C
and D. Label the intersection of the new arcs on the opposite side
as point E. Draw segments
 ,
 ,
 , and
 . By the
construction, PC = PD and EC = ED.
Now,
remember a locus theorem: The locus of points equidistant from two
points (C and D), is the perpendicular bisector of the line segment
determined by the two points. Hence,
 is the perpendicular
bisector of  .
The fact that we
created a bisector, as well as a perpendicular, is
actually MORE than we needed - we only needed to create a perpendicular.
Yea, free stuff!!!
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