CIRCLES

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CIRCLES
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Circles are all around us. They are products of
both our natural environment and of our
artificial environment.
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Circles are part of the astronomically big
and the infinitesimally small.
Throughout history circles have been considered
to have a certain perfection and harmony to them.
Even today there is a certain mystic and wonder
around the circle.
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CIRCLE TERMINOLOGY
To study circles, it is important that we are
familiar with the terminology of circles.
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A circle is defined by a group of points all
being the same distance from a point identified
as the centre, O.
A circle defines three distinct groups of points
- the points on the circle.
- the points inside the circle (including
centre).
- the points outside the circle.
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Radius a line segment drawn from the centre to
any point on the circle.
Chord a line segment between any two points on
a circle
Note A line segment is a line which has a
beginning and an end (the endpoints) and
consequently it has a defined length and can be
measured. When naming line segments, the two
endpoints are always used in either order.
O
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Diameter a chord that passes through the centre
of a circle.
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Arc a series of points on the circumference
between two given points on the circumference.
Arcs are classified into two groups

• Minor arcs arcs that make up less than one half
  of the circumference.

• Major arcs arcs that make up one half or more
  of the circumference.

When naming arcs,
Minors arcs can be named using 2 letters
representing the endpoints (although the 3 letter
designation can also be used as with major arcs).
3 letters must be used to name a major arc (two
letters to identify the ends of the arc and a
third letter between them to identify any other
point on the arc).
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Line a series of points along a straight line
that go infinitely in both directions.
To name a line you can use any two points on the
line
Secant a straight line that intersects a circle
at two points.
Tangent a straight line that intersects a
circle at one point.
Point of Tangency point of intersection which
is shared by the tangent line and the circle. T
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DISTINCTION BETWEEN CHORDS AND ARCS
Chords
Subtend
Arcs
However it does NOT make sense to say that
Apples Steal Criminals
In the same way it doesnt make sense to say that
Arcs Subtend Chords
subtends
therefore
but
does not subtend
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When referring to chords, radii, arcs, diameters,
secants and tangents, it is important to use the
correct symbol above the letter names.
- used for line segments (chords, radii or
diameters)
- used for lines (tangents or secants)
- used for arcs (minor or major)
B
The 3 geometric items above are all referring to
different sets of points.
- refers to all points on the circumference
between A and B
A
- refers to all points on a straight line between
A and B
- refers to all points on a straight line between
and beyond A and B
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Angles are also elements of a circle that are
often referred to. An angle is a rotational
separation between two rays. The angles of a
circle are classified into 4 categories based on
the location of their vertices
(1) Central angle vertex at the centre
(2) Interior angle vertex inside the circle
(3) Exterior angle vertex outside the circle
(4) Inscribed angle vertex on the
circumference
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?AOB is a central angle because its vertex, O, is
at the centre of the circle. Incidentally, it is
also an interior angle because its vertex is
inside the circle. However, just as we dont
refer to a square as a rectangle (eventhough it
is) so do we not refer to a central angle as an
interior angle.
-formed by two radii
?AOB intersects
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?DEF is an interior angle because its vertex, E,
is inside of the circle.
-formed by two chords
O
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?IKM, ?IKN and ?MKN are all exterior angles
because their vertex, K, is outside of the
circle.
-formed by two secants, two tangents or a secant
and a tangent
O
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?QPT, ?QPR and ?RPT are all inscribed angles
because their vertex, P, is on the circumference
of the circle.
-formed by two secants or a secant and a tangent
O
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final
The end
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