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Extension Geometric Constructions
B
A
C
EXTENSION
Geometric Constructions
Geometria Renaissance
sculptor Antonio de Pollaiolo
(1433 – 1498) depicted Geometry
in this bronze relief, one of ten
reliefs on the sloping base of the
bronze tomb of Sixtus IV, pope
from 1471 to 1484. The important
feature in this photo is the
inclusion of compasses, one of
two instruments allowed in
classical geometry for
constructions.
The Greeks did not study algebra as we do. To them geometry was the highest
expression of mathematical science; their geometry was an abstract subject. Any practical application resulting from their work was nice but held no great importance. To
the Greeks, a geometrical construction also needed abstract beauty. A construction
could not be polluted with such practical instruments as a ruler. The Greeks permitted
only two tools in geometrical construction: compasses for drawing circles and arcs of
circles, and a straightedge for drawing straight line segments. The straightedge, unlike
a ruler, could have no marks on it. It was not permitted to line up points by eye.
Here are four basic constructions. Their justifications are based on the congruence properties of Section 9.4.
Construction 1 Construct the perpendicular bisector of a given line segment.
Let the segment have endpoints A and B. Adjust the compasses for any radius
greater than half the length of AB. Place the point of the compasses at A and draw
an arc, then draw another arc of the same size at B. The line drawn through the points
of intersection of these two arcs is the desired perpendicular bisector. See Figure 23.
A
A
B
r
FIGURE 23
FIGURE 24
Construction 2 Construct a perpendicular from a point off a line to the line.
(1) Let A be the point, r the line. Place the point of the compasses at A and draw
an arc, cutting r in two points.
(2) Swing arcs of equal radius from each of the two points on r which were constructed in (1). The line drawn through the intersection of the two arcs and point A
is perpendicular to r. See Figure 24.
(continued)
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510
CHAPTER 9
Geometry
Construction 3 Construct a perpendicular to a line at some given point on the
line.
(1) Let r be the line and A the point. Using any convenient radius on the
compasses, place the compass point at A and swing arcs which intersect r, as in Figure 25.
r
A
FIGURE 25
Euclidean tools, the compasses
and unmarked straightedge,
proved to be sufficient for Greek
geometers to accomplish a great
number of geometric
constructions. Basic constructions
such as copying an angle,
constructing the perpendicular
bisector of a segment, and
bisecting an angle are easily
performed and verified.
There were, however, three
constructions that the Greeks
were not able to accomplish with
these tools. Now known as the
three famous problems of
antiquity, they are:
(2) Increase the radius of the compasses, place the point of the compasses on
the points obtained in (1) and draw arcs. A line through A and the intersection of the
two arcs is perpendicular to r. See Figure 26.
A
r
FIGURE 26
Construction 4 Copy an angle.
(1) In order to copy an angle ABC on line r, place the point of the compasses at
B and draw an arc. Then place the point of the compasses on r at some point P and
draw the same arc, as in Figure 27.
A
1.To trisect an arbitrary angle;
P
2.To construct the length of the
edge of a cube having twice the
volume of a given cube;
r′
r
3.To construct a square having
the same area as that of a
given circle.
B
C
FIGURE 27
In the nineteenth century it
was learned that these
constructions are, in fact,
impossible to accomplish with
Euclidean tools. Over the years
other methods have been devised
to accomplish them. For example,
trisecting an arbitrary angle can
be accomplished if one allows the
luxury of marking on the
straightedge! But this violates the
rules followed by the Greeks.
(2) Measure, with your compasses, the distance between the points where the
arc intersects the angle, and transfer this distance, as shown in Figure 28. Use a
straightedge to join P to the point of intersection. The angle is now copied.
A
r
B
C
r′
P
FIGURE 28
There are other basic constructions that can be found in books on plane geometry.
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Extension
Geometric Constructions
511
EXTENSION EXERCISES
Use Construction 1 to construct the perpendicular bisector of segment PQ.
1.
P
Q
Use Construction 3 to construct a perpendicular to the
line r at P.
5.
r
P
6. r
2.
P
P
Q
Use Construction 4 to copy the given angle.
7.
Use Construction 2 to construct a perpendicular from P
to the line r.
3. r
A
8.
P
P
9. Construct a 45 angle.
4.
r
10. It is impossible to trisect the general angle using only
Euclidean tools. Investigate this fact, and write a
short report on it. Include in your report information
on the construction tool called a tomahawk.
P
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