A circle is defined
by a point P, called the center, and a distance r, called the
radius, such that the set of points whose distance from the
point P is equal to the radius r.
Since most of the material
in the chapter on circles is review of high
school geometry, it is assumed that only a brief review is
needed.
Vocabulary with Circles
The following words are used with circles:
semicircle, secant, tangent, chord, diameter, radius,
center, central angle, and inscribed angle.
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Basic Circle
Theorems
Theorem
1: A circle and a line cannot
intersect in three or more points.
Theorem 2:
A line tangent to a circle is perpendicular to
the radius drawn to the point of tangency.
Theorem 3:
Two chords which intersect in a circle have the
proportion that the the product of the two segments of
one chord equals the product of the two segments of
the other chord.
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Angle
Measurement Theorems
What is important is
that all of these theorems are derived from the first
theorem.
Theorem 1:
A central angle is measured by the arc it
subtends.
Corollary 1: An
inscribed angle is measured by 1/2 the arc it
subtends.
Corollary 2: An
angle inscribed in a semicircle must be a right
angle.
Theorem 2: An
angle formed by two chords is measured by 1/2 the sum
of the two arcs it subtends.
Theorem 3: An
angle formed by two secants is meandered by 1/2 the
difference of the two arcs it subtends.
Corollary 3: An
angle formed by a tangent and a secant is measured by
1/2 the difference of the two arcs it subtends.
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Quadratic
Equation Solutions
Given a quadratic equation of the form:
x2 + ax +b2 = 0, a geometric
solution can be found which uses the fact that a= the
sum of the two factors and b2 = product of
the two factors.
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Sketchpad
Circles
Sketchpad has the circle tool
which allows circles to be drawn easily.
However, if you want to define a
particular circle, use:
1) Construct->Circle by
Center+Point
on selecting two points. Make sure
that you select the point to be the
center as your first point.
2) Construct->Circle by
Center+Radius
on selecting a point and a segment,
the latter being the radius. Here, the
order of selection does not matter.
Question:
Do three
points always determine a
circle?
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