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GEOMETRY – PRECISION
Measuring.
We have already spoken of geometry as an English word that comes from two
Greek words. GEO means earth or land. METRY means to measure a thing.
The ancient Greeks learned their earliest geometry from the even more ancient
Egyptians, who needed reliable ways to measure land. Egypt is a desert, with
practically no rainfall. This would make farming impossible were it not for the
presence of the Nile River flowing from central Africa north to the Mediterranean
Sea. Every year in the spring the Nile would flood, filling its valley with water and
mud. This was good, because it meant the Egyptians could grow a lot of food to
feed themselves through the year. But it was also a problem, because the mud
would bury the boundary lines of the farms.
The Egyptians found an elegant solution to this problem. They developed
techniques for measuring the land each year to redetermine the boundaries of
the farms. This was the origin of geometry, which meant “measuring land”.
Later, people realized that these same techniques could be used for other things.
Geometry became the basis for planning buildings and cities, for constructing
pyramids, temples, and fortresses.
Eventually the Greeks recognized the abstract principles behind these practical
techniques. They analyzed geometry into some basic truths that apply to the
world around us. We now get to study what they discovered.
The shapes we work with in geometry are called figures. To draw accurate
pictures of most of these figures we need only two simple tools. One is the
straight edge, so we can draw straight lines. The other is the compass, which we
use to mark off lengths of things.
When you want to draw a straight line you usually use a ruler. The ruler comes
with distance markers on it, typically in inches or centimeters. They can be used
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to measure approximate lengths, but they are not accurate enough for true
geometry. So, when you are trying to be precise in geometry, the ruler is used
only for its straight edge.
To measure a distance or a length in geometry, we use a different tool: the
compass. We use it because it is more accurate. For the rough geometry done by
the Egyptians, they might have used a piece of string or a rope to define a length.
They could lay out that measured length repeatedly in a straight line to define
two or three or more multiples of that length. However that string or rope might
shrink or stretch a little, reducing the accuracy of that measurement. Centuries
later, land surveyors learned to use chains, that do not stretch or shrink so much.
Modern surveyors use electronic laser devices to improve accuracy. When we do
geometry, we must try to achieve perfect measurements to fit the abstract idea
of perfect shapes.
The compass is a good (although not perfect) way to define short distances. It has
two legs that hinge on an axis. You spread the legs apart until the distance
between them is the one you want. Then you tighten a screw near the axis to
make that distance remain fixed on the compass.
This fixed distance is called a unit. It does not need to be a numbered amount like
3 inches, or 10 centimeters, or 4 miles. It is simply fixed as one unit in the task
being done at the moment. You can mark off a single unit in any direction. You
can mark off 2 units, or 3 units, or any multiple of that unit simply by laying them
out on a straight line. That is what the straight edge is for. This can be done on a
piece of paper, or on any flat surface.
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Drawing the circle.
The easiest shape to draw is the circle. Choose a length and set it on the
compass. This need not be a specific distance measurable in centimeters, or
inches, or anything else. It is a length chosen and set by you for your
convenience. While you are working with this figure (the shape you are drawing)
this length will be called one unit.
Now you are ready to draw a circle. Put a pencil dot on the paper where you
want the center of the circle to be. (This dot is not absolutely needed, but you
will probably find it useful later if you have marked that dot beforehand.) Now
put the sharp point leg of the compass on that dot (the center of the circle) and
touch the other leg of the compass (the one with the pencil lead) to the paper.
Then, holding the compass down carefully, twist the compass around the center
point. The pencil lead will mark a continuous curving line that gradually circles
around the center until it meets the place where it started. That closed figure is a
circle. (A really pretty circle.)
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This next illustration shows some of the details of what you have just done. The
spot labeled “center” is where you put the dot to place the sharp point of the
compass. The distance labeled “radius” is the length you set on the compass as
one unit in this figure. The curved line labeled “circumference” is the circle itself.
The center and the radius are not part of the circle, but they are essential
elements for creating the circle. Even if they were not in the illustration, you
would know that they are real.
Now let us look at a couple of interesting details about this circle. (These details
will be true of every circle.) Lay your straight edge on the paper very close to the
center point you marked ahead of time. Using a separate pencil, draw a straight
line from the center to any point on the edge (the circumference) of the circle.
That line is a radius. It will be exactly one unit long, because you drew the circle
in such a way that every point on the edge of the circle is exactly one unit from
the center.
Now, again lay your straight edge on the paper close to the center point, and
draw a radius to the edge of the circle. (You can reuse the radius you drew
before, or start a new one. It will not matter because every radius of this circle
will be exactly one unit long.) But this time be careful not to let the straight edge
move on the paper. Take that radius line you just drew and extend it on the other
side of the center point all the way to the opposite side of the circle. This line is
called a diameter (dia = two, meter = measure). It will be exactly two units long,
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because it is really two radii marked off next to each other on a single straight
line. The diameter is the longest possible straight line you can draw from any
point on the edge of the circle to a point somewhere else on the edge of the
circle. To be that longest possible line, it must pass through the center of the
circle.
The next illustration shows a diameter going from one edge of the circle to the far
side. Since it is a diameter it passes through the center of the circle. There is
another line called a chord. It goes from one point on the edge of the circle to
another point on the edge, but it does not pass through the center of the circle.
Every chord, no matter where you draw it, will be shorter than the diameter.
(There is also a line outside the circle called a tangent. It is a straight line that
touches the edge of the circle at one and only one point. We will not need to use
that line until much later.) There is also a part of the circumference of the circle
labeled “arc”. That word means any portion of the edge of the circle.
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Drawing equilateral triangles.
Let us see how to draw an equilateral triangle. (Remember, it has three sides of
equal length, and three equal angles.) First, draw a circle, and be sure to start by
marking its center point. (Actually, we do not need the whole circle this time,
only an arc of about one-quarter of it. But it will not hurt if you draw the whole
circle.) Then, using the straight edge, draw a radius from the center to some
point on the edge of the circle. Now move the sharp point of the compass to the
point where the radius meets the edge of the circle. Then twist the compass to
the point where the pencil end crosses the edge of the circle. Use the straight
edge to draw a line connecting those two points on the edge of the circle. What
you have drawn is a chord. That chord has the same length as the radius because
you used the preset distance on the compass to define both those lengths. Now
use the straight edge to draw a line from the center of the circle to the open end
of the chord. Since this new line is a radius it must be the same length as the
other two, the radius and the chord, which you have already drawn. What you
have made is a closed figure with three sides and three angles. That is a triangle.
And, since you know all three of its sides are of equal length, it must be an
equilateral triangle. And, since it is an equilateral triangle, its three angles must
be equal to each other. (Also, you can see now why you did not need the whole
circle to draw this figure: about one-quarter of the circle was more than enough.)
Let us start again. Mark a center for a circle, and use the compass to draw the
entire circle. (This time you will need the entire circle.) Now, following exactly
the procedure in the previous paragraph, draw in an equilateral triangle. Then,
from an outer point of the triangle, use the compass to measure another radius
length chord out at the edge of the circle. Use the straight edge to draw that
chord, and from its open end draw another radius back to the center of the circle.
That gives you two identical equilateral triangles, with one shared line between
them. Repeat the procedure to draw a third equilateral triangle which shares one
line with a previous triangle. Continue, using the same procedure to draw a
fourth, fifth, and sixth triangle. The last triangle should be the easiest, because
two of its sides are already there, and all you have to draw in is the chord. These
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six equilateral triangles should fit exactly in the one circle, if you drew them with
sufficient care.
Look at the beautiful figure you have just drawn: six equilateral triangles filling a
circle. From the center there are six radii drawn out to the edge of the circle. If
you lay your straight edge on any one of those radii, you will see that continuing
on the other side of that radius is another perfectly aligned with it. Those two
radii lying on a single straight line constitute a diameter, which by definition is the
longest possible straight line from one edge of the circle to the opposite edge. It
must pass through the center of the circle, and it must be exactly the length of
two radii combined. There are six of these diameters in the figure, although it
may seem like only three since each duplicates another drawn from the opposite
edge of the circle. On each side of any one of those diameters you will find
exactly three of your equilateral triangles. This elegant symetry is one of the
amazing things about geometry. The ancient Egyptians thought they were just
measuring land, but when you look into the subject more deeply, you uncover
deep and inescapable truths.
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Drawing more triangles.
Start again by drawing a circle with the center marked. Now draw a radius. Then
draw a chord chosen to be shorter than the radius. And draw another radius from
the free end of the chord back to the center of the circle. You have made a
triangle, with two sides equal, but the third side not equal. That is an isoceles
triangle.
Now go through the same procedure, but tjhis time make the chord be longer
than a radius. Again you have made an isoceles triangle, but in this one the
nonmatching side is the longest rather than the shortest side. And remember
that in each of these last two cases, the angles opposite the equal sides are equal
angles.
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We know that it is possible to have triangles in which none of the sides are equal.
They are called scalene triangles. You can draw a scalene triangle with sides of
three arbitrary lengths. Just be careful that you do not accidentally draw an
isoceles or equalateral triangle if you really want a scalene triangle. Also,
remember that the sum of the lengths of the two shorter sides must be greater
than the length of the longest side. It will also be true of a scalene triangle that
none of the angles are equal, precisely because none of the opposite sides are
equal.
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Drawing parallelograms.
Step 1. Draw a line. (That is what the straight edge is for.) Then choose a
distance on the compass, and draw an arc of a circle from a point on the line. Pick
another point on the line and draw another arc with the same radius as the first
arc. (This means you were careful not to let the distance defined on the compass
change.)
Step 1:
Step 2: Now use the straight edge to draw the line that just touches those two
arcs but does not cross them. You have just drawn parallel lines. The shortest
distance from the first line to the second line is the same no matter where you
start on either line.
Step 2:
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Step 3: Draw another another line that crosses the two parallel lines you just
drew.
Step 3:
Step 4. It may help if you erase the lines you no longer need.
Step4:
Step 5. Choose a distance to be measured by the compass, and mark the distance
on line one, using the intersection of lines one and three as the center point.
Keeping that same distance on the compass, and using the intersection point of
lines two and three as the new center point, mark this distance with an arc on line
two. (Be sure to work on the same side of line three. You want a closed figure,
not an “X”.)
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Step 5:
Step 6. Use the straight edge to connect the points where the arcs are marked on
lines one and two. This fourth line, and the earlier third line are parallel, because
the shortest distances between three and four are always the same. (Notice that
this shortest distance is usually not the same as the compass distance you used to
mark the arcs form line three on lines one and two. If the shortest distance from
line three to line four is the same as the arc distance you used, then the figure you
have just drawn is a rectangle.) In any case, you have just made a parallelogram.
This is a closed figure made by two pairs of parallel lines that intersect at four
points, making four angles. The pairs of opposite angles will be equal.
Step 6:
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Step 7. Again, it may help if you erase the lines you no longer need.
Step 7:
It was a lot of work, but you should be proud of the beautiful parallaeogram you
have drawn.
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Drawing right angles
We have spoken of three kinds of angles. Acute angles are smaller (sharper) than
right angles. Right angles are correct (perfect), like a corner of a square. And
obtuse angles are larger (blunter) than right angles. Obviously right angles are the
standard against which all angles are measured. Right angles do have many
special uses.
We can draw a right angle using only a compass and a straight edge, just as we
have drawn all our other figures. To start, we need two circles that overlap a bit
(or at least the centers of those two circles and arcs of those two circles that
include the two places where they overlap). We use a straight edge to connect
the centers of those two circles. (The circles do not even have to be the same
size. They just have to overlap.) Then we use the straight edge to draw a line
connecting the two points of intersection where the two circles overlap. (There
will be exactly two intersection points provided we did not put the centers of the
two circles too close together or too far apart.)
These two straight lines, the one connecting the centers of the two circles, and
the other connecting the two points where the circles intersect, will cross each
othe at exactly one point. At that point the two crossing lines will make four
angles. Each of those four angles will be a right angle. Since all four will be right
angles, they will all be the same size, they will all equal each other.
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Right angles:
Now that we can draw a right angle, we can use it to make right triangles. We can
also use the right angle to make rectangles. To draw a rectangle, we simply
follow the rules for making a parallelogram, but be sure that one of its angles is a
right angle. If we do this correctly, that one right angle in the parallelogram will
force all its angles to be right angles. We will have achieved a closed figure with
four right angles formed by two pairs of parallel lines. That is the definition of a
rectangle. If, in addition, we are careful to make the lengths of all the sides equal,
our figure will fit the definition of a square. And we can do all this with just a
compass and a straight edge!
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Bisecting a line:
If, when drawing right angles as in the previous procedure, we had made the two
circles equal (matching radii), the two straight lines involved would have each
been cut exactly in half. You could call it a multi-purpose procedure.
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