Geometry Lesson
Chapter 1
Lesson 1
Draw a circle.
The circle refers to the boundry. The inside of the circle is called a
disk.
When we say area of circle, we really mean the area enclosed by the
circle, or the area of the disk.
The center of the circle is also the center of the disk.
Draw a horizontal line segment through the center of the circle.
A line segment is a finite part of a line.
A line is infinitely long.
Consider the line that contains the line segment you just drew.
This line is called the x-axis.
Draw a vertical line segment through the center of the circle.
This line containing that line segment is called the y-axis.
The center of the circle is called the origin.
Note that you have cut the circle into 4 parts.
The upper right part is called the 1st quadrant.
The upper left part is called the 2nd quadrant.
The lower left part is called the 3rd quadrant.
The lower right part is called the 4th quadrant.
Pick a point on the circle in the first quadrant.
Draw a line segment from the origin to that point.
The line segment from the center of the circle to a point on the circle is
called a radius.
Draw a line segment from that point straight down to the x-axis.
Draw a line segment from where you met the x-axis back to the origin.
You have just drawn a right triangle.
It is called a right triangle because one of its angles is a right angle.
A right angle is an angle between perpendicular lines.
The longest side of a right triangle is called the hypotenuse. The other
two sides are called the legs of the right triangle.
The triangle has three vertices. They are the origin, the point on the
circle that you drew the first line segment to, and the point on the
x-axis just below that point.
An equilateral triangle is a triangle with all three sides equal.
An isosceles triangle is a triangle with at least two sides equal.
An equilateral triangle is also an isosceles triangle.
In an isosceles triangle with one of the sides a different length, the
side of different length is called a base. The sides that are of equal
length are called the arms.
In an equilateral triangle any of the sides may be chosen as base.
A scalene triangle has all three sides of different length. No two sides
are equal.
Also, in a scalene triangle, any of the sides may be chosen as base.
A line segment through the center of the circle from one point of the
circle to its opposite point is called a diameter.
The diameter is twice the radius.
A part or segment of a circle is called an arc.
An arc half way around the circle is called a semicircle.
Draw another circle inside or outside of the first circle you drew, and
make it have the same center.
You have just drawn two concentric circles.
How many different kinds of triangles can you draw within the circle?
Draw some triangles
(1) with origin as one vertex, a point of the circle as another vertex,
and a point on the x-axis just below the circle as a third vertex.
(2) with the antipodal points at opposite ends of a diameter as two
vertices, and some other point of the circle as the third vertex.
(3) with the origin as one vertex, and points on the circle as the other
two vertices.
The opening between the sides of the triangle are called angles.
A triangle has 3 angles.
The name triangle means it has 3 angles.
From each vertex of a triangle draw two circles with that vertex as
center. Make the radius of one circle the same as the longest side from
that vertex, and make the radius of the other circle the same as the
shortest side from that vertex.
Now draw a line segment on the x-axis.
Draw two circles, centered at each endpoint of the line segment, such that
the circles do not intersect.
Draw another circle centered at one of the endpoints such that the
two previously drawn circles are inside it.
What do these circles tell you about how long two sides of a triangle may
be if the length of one side has already been chosen?
Whenever we draw a line on paper, we are really drawing a line segment.
Most of the time it doesn't matter whether we think of the line segment we
draw as a line segment or as a line. But in those cases when we need to
make clear that we mean a line rather than a line segment, then we can
draw the line with an arrow head at each end.
<-------------------------> represents a line. Both ends are at
infinity.
------------------------> represents a ray. A ray has one end where you
can see it and the other end at infinity.
---------------- represents a line segment.
Also it is customary to use the word line for both line and line segment.
We will depend on context to show whether we mean the infinite length line
or the finite length line segment.
For example, when we draw a triangle, we may speak of the three lines
making up the sides of the triangle. It is obvious that we really mean
three line segments that make up the sides of the triangle.
To draw line segments we use a ruler or straightedge. If you use the
ruler as a straightedge, ignore the measuring marks on the ruler. In
classical plane geometry, rulers were not available. Straightedges are
what the ancient greeks used, and the plane geometry you are learning is
very close to the geometry taught in Ancient Greece.
The straightedge allows us to construct a straight line given two points.
This construction corresponds to Euclid's first axiom. Two points
determine a line. Euclid thought of this axiom as an obviously true
statement because with a straightedge he could draw a line segment between
any two preassigned points.
Notice that I said construct a straight line instead of draw a straight
line. When we use the compass to make a circle or a straight edge to make
a line, then we say construct the circle or construct the line rather than
say draw the circle or draw the line.
Because the teaching of plane geometry is based on the plane geometry
taught in Ancient Greece, it is useful to have the word "construct" to
mean drawn by using ONLY the straightedge and compass.
The first fundamental construction to show you is how to make a line
segment equal in length to a given line segment.
A-----------------------B
Suppose I wish to construct a line segment equal in length to the line
segment
A-----------------------B.
1. Use your straightedge to construct a line
C----------------------------------------D
clearly longer than the given line.
2. Place the metal point of the compass on A and the pencil point on B.
3. Tighten the compass so that when you lift it off the paper, it remains
open the distance between A and B.
4. Place the metal point of the compass on C and with the pencil point,
draw a short arc across CD. Give the name E to the place where the arc
cross the line CD.
C-----------------------)-----------------D
E
The line segment CE is the same length as AB.
You can use this basic construction to add line segments.
Suppose you wanted to add the line sebments
A-----------------------B
and the line segment
C------------D
Then construct a copy of AB on another line. Then at the endpoint of that
copy, construct a copy of CD.
E------------------------)-------------)---------------F
G H
EG has been constructed to be the same length as AB and GH has been
constructed as the same length as CD.
EH is now the line segment AB plus the line segment CD. Line segments
are line numbers. We can add the just as we add numbers. And we can
subtract a smaller line segment from a larger line segment.
Since lengths are positive numbers we cannot subtract a larger line
segment from a smaller line segment. If we wish to represent both
positive numbers and negative numbers by line segments, then we use
directed line segments, also called vectors. I will talk more about
vectors in a later lesson.
The second fundamental construction is to make an equilateral triangle
with all sides equal to a given line segment.
A--------------B
We will construct an equilateral triangle with each side the same length
as AB above.
1. Place the metal point of your compass on A and the pencil point on B.
Another way to say the same thing is to say: Make A the center for your
compass, and make the length AB the radius.
Make an arc with the compass above the line segment AB about half way
between A and B.
2. Make B the center for the compass, and with the same AB radius, make
another arc above the line segment AB. Make the second arc intersect the
first. If the arcs do not intersect, go back and construct the first arc
longer so that you can make them intersect.
3. Label the point of intersection C.
Using the straightedge, connect the point A to C and the point B to C.
You have now constructed the required equilateral triangle.
The third fundamental contruction is to bisect a given line.
To bisect a line means to cut it into two equal parts.
The point where we cut the line into two equal parts is called the
midpoint.
Let's bisect the line
A--------------------------------B.
1. With A as center and AB as radius, construct an arc on each side of
AB.
2. With B as center and AB as radius, construct an arc on each side of AB
that intersects the arcs drawn in step 1. If the arcs do not intersect,
start over with a larger radius. The radius should be greater than half
the distance from A to B.
3. Label the points of intersection C and D.
4. With the straightedge, connedt the line C to D. The line segment
connecting C to D will bisect the line segment AB. Label the point where
CD intersects AB with the name E.
The line segment AB is twice the line segment AE.
And AE is same length as EB. And AE + EB = AB.
Practice Exercies.
1. Draw two short line segments AB and CD. On a working line, construct
a line equal in length to 2 * AB + 3 * CD.
2. Draw two line segments AB and CD such that AB is longer than CD.
Construct AB - CD.
3. Using your compass, draw 3 concentric circles. Concentric circles
are circles with the same center and different radii.
4. Construct an isoceles triangle with base a different length than the
arms.
5. Construct a Scalene triangle.
6. Draw 3 line segments at different 3 different angles. Construct the
bisection of each of those line segments.
7. Construct a triangle. Then bisect one of the sides of the triangle.
8. Construct a triangle. Then bisect two of the sides of the triangle.
9. construct at triangle. Then bisect all three of the sides of the
triangle.
10. If the base of an isoseles triangle is 4 inches and each of the arms
is 6 inches, what is its perimeter?
11. Fill in the following table.
Lengths of sides of triangle
side 1 side 2 side 3 perimeter
3 4 5
5 12 13
1 2 1
1 2 1.2
1 1 1.5
4 4 4
6 a a 24 Note, the two missing sides represented
by the letter a are the same length.
3 4 12
a a a 21
a a a 63
12. If you know the perimeter and the length of the base of an isoceles
triangle, how do you find the length of one of the two equal arms?
13. If you know the perimeter of an equilateral triangle, how do you find
the length of one side?
14. Construct a scalene triangle.
Bisect each of the three sides.
Connect the three middle points by straight lines.
15. Construct a line segment AB. Bisect the line segment AB with line
segment CD using the line bisection construction. The segment CD is
called the perpendicular bisector of AB because CD bisects AB and because
CD is perpendicular to AB.
Label the point of bisection with the letter F.
Using successively, the points A, B, C, D and centers and with the radius
AF, construct the circles. That is, Construct 4 circles all with radius
AF about the 4 centers, A, B, C, D.
16. Construct an equilateral triangle. Label the vertices A,B,C. Bisect
each side. Label the points of bisection, D,E,F. Make D on the line
connecting B and C. Make E on the line connection A and C. Make F on the
line connecting A and B.
Put the metal point of the compass on vertex A of the triangle. Put the
pencil point on point E. Tighten the compass to fixe the radius to be AE.
With A as center construct the circle through E and F.
With B as center construct the circle through D and F.
With C as center consturct the circle through D and E.
17. Construct an Equilateral triangle. Label the vertices A, B, and C.
Bisect AB. Extend the bisection of AB until it crosses the point C.