幾何尺規作圖
Geometric Construction By Compasses and Straightedges
F.1 ( )
Name: _________________(
)
The word construction in geometry has a very specific meaning: the drawing of geometric items such as
lines and circles using only a compass and straightedge or ruler. Very importantly, you are not allowed to
measure angles with a protractor, or measure lengths with a ruler.
The compass (圓規) is a drawing instrument used for drawing
circles and arcs. It has two legs, one with a point and the other
with a pencil or lead. You can adjust the distance between the
point and the pencil and that setting will remain until you
change it.
A straightedge (直尺) is simply a guide for the pencil when drawing
straight lines. In most cases you will use a ruler for this, since it is
the most likely to be available, but you must not use the markings on
the ruler during constructions.
The ancient Greek mathematician Euclid (歐幾里德) is the acknowledged
inventor of geometry. He did this over 2000 years ago, and his book
"Elements"(《幾何原本》) is still regarded as the ultimate geometry
reference. In that work, he uses these construction techniques extensively,
and so they have become a part of the geometry field of study. They also
provide a greater insight into geometric concepts and give us tools to draw
things when direct measurement is not appropriate.
A few points to remember when doing the types of geometric constructions covered in these lessons:
* Do not use a protractor
* Use a sharp pencil
* Do not erase the construction lines.
(They will show the teacher how the lines or shapes were constructed)
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Basic Geometric Constructions:
1. Given point P on line k, construct a line passing through P and perpendicular to k.
(過線上一點作一直線垂直於一已知直線)
a. Begin with line k, containing point P.
k
P
b. Place the compass on point P. Using an arbitrary radius,
draw arcs intersecting line k at two points. Label the
intersection points X and Y.
c. Place the compass at point X. Adjust the compass radius
1
so that it is more than XY. Draw an arc as shown
2
here.
k
X
P
Y
P
Y
k
X
d. Without changing the compass radius, place the
compass on point Y. Draw an arc intersecting the
previously drawn arc. Label the intersection point A.
A
k
X
Y
P
e. Use the straightedge to draw line AP. Line AP is
perpendicular to line k.
A
k
X
P
Y
Exercise: Construct a line passing through Q and perpendicular to L.
L
L
+
Q
Q+
2
2. Given point R, not on line k, construct a line through R, perpendicular to k.
(過線外一點作一直線垂直於一已知直線)
a. Begin with point line k and point R, not on the line.
R
b. Place the compass on point R. Using an arbitrary radius,
draw arcs intersecting line k at two points. Label the
intersection points X and Y.
R
c. Place the compass at point X. Adjust the compass radius
1
so that it is more than XY. Draw an arc as shown
2
here.
d. Without changing the compass radius, place the
compass on point Y. Draw an arc intersecting the
previously drawn arc. Label the intersection point B.
k
k
Y
X
R
k
Y
X
R
k
Y
X
B
e. Use the straightedge to draw line RB. Line RB is
perpendicular to line k.
R
Y
X
B
Exercise : Construct a line passing through P and perpendicular to L.
L
L
P+
P+
3
k
3. Construct the perpendicular bisector of a line segment. (作中垂線)
Or, construct the midpoint of a line segment.
a. Begin with line segment XY.
Y
X
b. Place the compass at point X. Adjust the compass radius
1
so that it is more than XY. Draw two arcs as shown
2
here.
c. Without changing the compass radius, place the
compass on point Y. Draw two arcs intersecting the
previously drawn arcs. Label the intersection points
A and B.
X
Y
A
X
Y
B
d. Using the straightedge, draw line AB. Label the
intersection point M. Point M is the midpoint of line
segment XY, and line AB is perpendicular to line
segment XY.
A
M
X
Y
B
Exercise : (a). Construct the perpendicular bisector of AB.
(b). Construct the perpendicular bisector of AC.
C
B
A
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4. Construct the angle bisector of an angle.(作角平分線)
a. Let point P be the vertex of the angle. Place the
compass on point P and draw an arc across both sides
of the angle. Label the intersection points Q and R.
Q
P
R
b. Place the compass on point Q and draw an arc across
the interior of the angle.
Q
P
R
c. Without changing the radius of the compass, place it
on point R and draw an arc intersecting the one drawn
in the previous step. Label the intersection point W.
Q
W
P
R
d. Using the straightedge, draw ray PW. This is the
bisector of QPR.
Q
W
P
R
Exercise : (a). Construct the angle bisector of ACB .
(b). Construct the angle bisector of CAB .
A
C
B
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5. Construct an angle congruent to a given angle. (作等角)
a. To draw an angle congruent to A,
begin by drawing a ray with
endpoint D.
A
D
b. Place the compass on point A and
draw an arc across both sides of the
angle. Without changing the
compass radius, place the compass
on point D and draw a long arc
crossing the ray. Label the three
intersection points as shown.
C
B
A
D
c. Set the compass so that its radius is
BC. Place the compass on point E
and draw an arc intersecting the one
drawn in the previous step. Label
the intersection point F.
E
C
F
B
A
D
E
d. Use the straightedge to draw ray
DF. Then EDF BAC.
C
F
B
A
D
Exercise : Construct PQR  AOC  BOD .
B
A
+
Q
D
O
C
6
E
6. Given a line and a point, construct a line passing through the point and parallel to the given line.
(過線外一點作一直線平行於一已知直線)
a. Begin with point P and line k.
P
k
b. Draw an arbitrary line through point P,
intersecting line k. Call the intersection point Q.
Now the task is to construct an angle with vertex
P, congruent to the angle of intersection.
P
k
Q
c. Center the compass at point Q and draw an arc
intersecting both lines. Without changing the
radius of the compass, center it at point P and
draw another arc.
P
k
Q
d. Set the compass radius to the distance between
the two intersection points of the first arc. Now
center the compass at the point where the second
arc intersects line PQ. Mark the arc intersection
point R.
P
R
k
Q
e. Line PR is parallel to line k.
P
R
k
Q
Exercise : Construct a line passing through the point Q and parallel to the given line L.
Q+
L
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7. Given a line segment as one side, construct an equilateral triangle. (作等邊三角形)
This method may also be used to construct a 60 angle.
a. Begin with line segment TU.
T
U
T
U
b. Center the compass at point T, and set the
compass radius to TU. Draw an arc as shown.
c. Keeping the same radius, center the compass at
point U and draw another arc intersecting the
first one. Let point V be the point of
intersection.
V
U
T
d. Draw line segments TV and UV. Triangle TUV
is an equilateral triangle, and each of its interior
angles has a measure of 60.
V
U
T
Exercise: Construct two equilateral triangles ABC and PQR .
+B
P
A+
8
R
8. Construct a regular pentagon. (作正五邊形)
a. Draw a circle and its diameter.
B
A
C
b. Draw the perpendicular bisector of AB.
A
B
D
c. Construct the perpendicular bisector C’D’, and
locate the mid-point M.
C
C’
M
A
B
D’
D
C
d. Using M as centre and MC as radius, draw an arc
to cut AB at E. CE is then the length of a side of
the required pentagon.
E
M
A
B
D
e. Using CE as radius and B as the first centre,
locate the points P, Q, R and S. Join BP, PQ, QR,
RS and SA to get a regular pentagon.
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