The Geometry of a Circle - By: Dennis Kapatos

Words in () are what the students should be
guided to conjecture. They should not be told
to them.
Teacher’s Instructions
Instructions (say aloud):
Construct a circle AB
Construct line AC through the center of circle AB;
hold down shift to make it vertical
Construct point D on line AC
Construct ray DE and ray DF as shown
Hide points B and C and line AC
Construct segment DA
Construct point G, the intersection of segment DA and
circle A
Swing rays DE and DF till they exactly touch the
circle, like you did with the string
Construct radius AH and AI to the points where it
looks like the rays touch the circle, make sure
H and I are on the rays, not the circle
Questions:
Does this seem like a very accurate way of
constructing this? What do you notice about the
radii and rays? (they almost look perpendicular)
Check this, measure angle AID and angle AHD,
what did people get? Let’s explore this further
Instructions:
Construct a new circle to the right
Construct a radius
Select it and its endpoint and construct a perpendicular line
Questions:
What happens when you move the point around? This line is called a tangent because it
intersects the circle at exactly one point. That point is called the point of tangency. What
should our definition of a tangent be? This is Theorem 12.2 and 12.3. From any one
point outside the circle, what how many different tangents can be drawn to one circle?
Instructions:
Construct a line through the center of the circle like you did for the other circle
Double click it to mark it as a mirror line
Select the point of tangency, the tangent, and the radius and reflect them
Hide the mirror line and it’s point
Construct the intersection of these 2 tangents.
Hide the 2 tangent lines
Construct the tangent segments
Construct the center segment and the point where it intersects the circle as
shown
Move the first point of tangency that you made around
Questions:
What do you notice about the lengths of the two tangent segments? (they’re
always the same length) Check this, measure them. Now move them around.
does this conjecture check? This is Theorem 12.4. Now getting back to our
question, we know all we need to in order to solve this question.
E
B
C
D
A
m ! A H D = 92.06 ° G
m ! A ID = 93.00 °
E
H
D
A
V
I
W
M
F
F
U
O