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Remember
-- use your compass and straight edge only! |
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Bisect
- cut into two congruent (equal) pieces.
Given:
Line segment AB
Task: Bisect line segment AB.
Directions:
1. Place your compass point on A and stretch the compass MORE
THAN half way to point B.
2. With this length, swing a large arc that will go BOTH above and
below segment AB. (If you do not wish to make a "large"
arc, you may simply place one small arc above AB and one small arc below
AB.)
3. Without changing the span on the compass, place the compass point
on B and swing the arc again. The two arcs you have now made should
intersect.
4. With your straightedge, connect the two points of intersection.
5. This new straight line bisects segment AB. Label the point
where the new line and AB cross as C.
Segment AB
has now been bisected and AC = CB. (It could also be said that they
are congruent.)
(I
like to use the "large arc method" because it creates a
"crayfish" looking creature which students easily remember and
which reinforces the circle concept needed in the explanation of the
construction.)
Explanation
of construction:
To understand the explanation you will need to label the point of
intersection of the arcs above segment AB as D and below segment AB as
E. Draw segments AD, AE, BD and BE. All four of these segments
are of the same length as they are actually radii of the same
circle. More specifically, DA = DB and EA = EB.
Now, remember a locus theorem: The locus of points equidistant from
two points, is the perpendicular bisector of the line segment determined
by the two points. Hence, DE is the perpendicular bisector of
AB.
This is
actually MORE than we needed - we only needed to create a bisector.
Isn't this great!
Given:
angle BAC
Task: Bisect angle BAC.
Directions:
1. Place the point of the compass on the vertex of angle BAC
(point A).
2. Stretch the compass to any length so long as it stays ON the
angle.
3. Swing an arc with the pencil that crosses both sides of angle
ABC. This will create two intersection points with the sides of
the angle.
4. Place the point on one of these intersection points
created on the sides of the angle BAC. If needed, stretch your
compass to a sufficient length to place your pencil well into the interior
of the angle. Stay between the sides (rays) of the angle.
Place an arc in this interior - you do not need to cross the sides of the
angle.
5. Without changing the width of the compass, place the point of the
compass on the other intersection point on the side of the angle and make
the same arc. Your two small arcs in the interior of the angle
should be crossing.
6. Connect the point where the two small arcs cross to the vertex A
of the angle.
You have now created
two new angles that are of equal measure (and are each 1/2 the measure of
angle BAC.)
Explanation of
construction:
To
understand the explanation, some additional labeling will be needed.
Label the point where the arc crosses side AB as D. Label the point
where the arc crosses side AC as E. And label the intersection of
the two small arcs in the interior as F. Draw segments DF and EF.
By the construction, AD = AE (radii of same circle) and DF = EF (arcs of
equal length). Of course AF = AF. All of these sets of equal
segments are also congruent. We have congruent triangles by SSS.
Since the triangles are congruent, any of their leftover corresponding
parts are congruent which makes angle BAF equal (or congruent) to angle
CAF.
Click
here to learn how to
Construct Perpendiculars On and
Off a Line
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