Analyzing Circles

1
Analyzing Circles

• OBJECTIVES
• Degree linear measure of arcs
• Measures of angles in circles
• Properties of chords, tangents, secants
• Equations of circles

2
About Circles

• Definition set of coplanar points equidistant
  from a given point P(center) ? written P
  
• Chord any segment having endpoints on the circle
• Radius (r) a segment from a point on the circle
  to the center
• Diameter (d) chord containing the center of the
  circle
• Circumference the distance around the circle
• Circumference C pd 2pr
• Concentric circles share the same center have
  different radius lengths

3
Angles and Arcs Measure

• Central angles have the vertex at the center of
  the circle
• The sum of non-overlapping central angles 360
• A central angle splits the circle into 2 arcs
• minor arc m
• major arc m
• Adjacent arcs share only the same radius
• The measure of 2 adjacent arcs can be
  added to form
• one bigger arc.

T
V .
P
L

• Arc Length is the proportion of the circumference
  formed by the central angle

4
Arcs and Chords
? chord
arc of the chord ?

• -Two minor arcs are iff their corr chords
  are
• - Inscribed polygons has each vertex on the
  circle
• - If the diameter of a circle is perpendicular to
  a chord, it bisects the cord the arc
• -Two chords are iff they are equidistant from
  the center.

11
11
.
5
Inscribed Angles
? Inscribed
Intercepted arc?

• An inscribed has its vertex on the circle
• Inscribed polygons have all vertices on the
  circle
• Opposite s of inscribed quadrilaterals are
  supplementary

• The measure of inscribed s ½ intercepted
  arc
• If an inscribed intercepts a semicircle, the
  90
• If 2 inscribed s intercept the same arc, the
  s are

?red blue s are
6
Tangents

• Tangent lines intersect the circle at 1 pointthe
  point of tangency

• A line is tangent to the circle iff it is
  perpendicular the the radius drawn at that
  particular point

• if a point is outside the circle 2 tangent
  segments are drawn from it, the 2 segments are
  congruent.

.
Tangents can be internal or external

?
?
7
Secants, Tangents Angle Measures
I

• A secant line intersects the circle in 2 points

intersecting at point of tangency
A
B
C
D
Central angles 1 secant 1 tangent
8
Secants, Tangents Angle Measures
II
intersection in interior of circle
B
C
1
2
A
D

• 2 secants forms 2 pair of vertical angles
• vertical

9
Secants, Tangents Angle Measures
III
Intersection at exterior point

• Case 1? 2 secants

C
B
P
A
D
C
D
Case 2? 1 secant 1 tangent
P
A
B
Case 3? 2 tangents
B
P
Q
A
10
Special Segments in a Circle
c
b
a

• If two chords intersect inside (or outside) of a
  circle, the products of their segments are
  equal ab cd

d
x

• 2 secants exterior point
• a(a x) b(b c)

a
b
c
1 tan and 1 sec exterior point
a
a2 x(x b) x2 bx
b
x
11
Equations of circles

• Point P (h, k) is the center of a circle.
• Radius of the circle r

y
(h, k)
x
The equation of this circle (x h)2
(y k )2 r 2