Apply properties of arc and chords

1. 6.14.1 Arcs and Chords
The student is able to (I can):
• Apply properties of arcs
• Apply properties of chords

2. circle The set of all points in a plane that are a
fixed distance from a point, called the
center.
A circle is named by the symbol and its
center.
AAAA
•
A

3. diameter
radius
central angle
A line segment whose endpoints are on the
circle and includes the center of the circle.
A line segment which has one endpoint on
the circle and the other on the center of
the circle.
An angle whose vertex is on the center of
the circle, and whose sides intersect the
circle.
A
•
C
B
D
CD is a diameter
AB is a radius
∠BAD is a central
angle

4. secant
tangent
point of
tangency
A line that intersects a circle at two points
A line in the same plane as a circle that
intersects it at exactly one point.
The point where the tangent and a circle
intersect.
•
A
B
m
C
chord
secant
tangent
point of
tangency

5. common
tangent
A line that is tangent to two circles.
common external tangent
common internal tangent

6. Theorem
Theorem
If a line is tangent to a circle, then it is
perpendicular to the radius drawn to the
point of tangency.
If a line is perpendicular to a radius at a
point on the circle, then it is tangent to the
circle.
BELine t ⊥
Line t tangent
to ⊙B •B
E
t

7. Theorem If two segments are tangent to a circle
from the same external point, then the two
segments are congruent.
•
S
A
N
D
SD ND≅

8. Examples The segments in each figure are tangent.
Find the value of each variable.
1.
2.
•
2a + 4
5a — 32
•
6y2 18y
5a — 32 = 2a + 4
3a = 36
a = 12
6y2 = 18y
y y
6y = 18
y = 3

9. minor arc
major arc
semicircle
An arc created by a central angle less than
180˚. It can be named with 2 or 3 letters.
An arc created by a central angle greater
than 180˚. Named with 3 letters.
An arc created by a diameter. (= 180˚)
•
S
P
A
T
SP (or PS)
is a minor arc.
SAP (or PAS)
is a
major
arc.SPA is a semicircle.
•
•
•

10. Examples Find the measure of each
1. = 135˚
2. = 360 — 135
= 225˚
3. m∠CAT
•
•
135˚
F
R
E
D
mRE
mEFR
C
A
T
260˚
= 360 — 260
= 100˚

11. If a radius or diameter is perpendicular to a
chord, then it bisects the chord and its
arc.
ER GO⊥
•
G
E
O
R
A
GA AO≅
≅GR RO

12. Example Find the length of BU.
•
B
L
U
E
3
2
5
2 2 2
3 x 5+ =
x
x = 4
BU = 2(4) = 8