8-3 & 8-4
TANGENTS,
ARCS & CHORDS
5 Theorems
THEOREM 1:
a line that intersects a circle is
tangent to a circle IFF it is
perpendicular to the radius drawn to
the point of tangency.
THEOREM 2:
If two segments from the same
exterior point are tangent to a circle,
then they are congruent.
EXAMPLES: SOLVE.
(Segments that appear to be tangent are.)
A
1.8 cm
B
C
FIND DC
F
7.0 cm
2.4 cm D
FIND CE and EA
E
Theorem 3:
In a circle, two chords are congruent iff their corresponding
minor arcs are congruent.
A
If AB  CD then AB  CD
E
C
Example:
B
D
Given mAB  127 , find the mCD.
Since m AB  mCD
mCD  127
Lesson 8-4: Arcs and Chords
5
Theorem 4:
In a circle, if a diameter (or radius) is perpendicular to a
chord, then it bisects the chord and its arc.
If DC  AB then DC bi sec ts AB and AB.
 AE  BE and AC  BC
D
Example: If AB = 5 cm, find AE.
If mAB  120 , find mAC.
AE 
A
E
B
C
AB
5
 AE   2.5 cm
2
2
m AB
120
m AC 
,  m AC 
 60
2
2
Lesson 8-4: Arcs and Chords
6
Theorem 5:
In a circle, two chords are congruent if and only if
they are equidistant from the center.
D
F
CD  AB iff OF  OE
Example:
If AB = 5 cm, find CD.
C
O
A
E
B
Since AB = CD, CD = 5 cm.
Lesson 8-4: Arcs and Chords
7
Try Some Sketches:
 Draw a circle with a chord that is 15 inches long and 8 inches from
the center of the circle.
 Draw a radius so that it forms a right triangle.
 How could you find the length of the radius?
Solution: ∆ODB is a right triangle and OD bi sec ts AB
AB 15
DB=
= =7.5 cm
2
2
OD=8 cm
OB2 =OD 2 +DB2
OB2 =82 +(7.5) 2 =64+56.25=120.25
OB= 120.25  11cm
Lesson 8-4: Arcs and Chords
A
15cm
D
8cm
B
x
O
8
Try Some Sketches:
 Draw a circle with a diameter that is 20 cm long.
 Draw another chord (parallel to the diameter) that is 14cm long.
 Find the distance from the smaller chord to the center of the circle.
AB 14

 7cm
Solution: OE bi sec ts AB.  EB 
2
2
14 cm
∆EOB is a right triangle. OB (radius) = 10 cm A
B
E
OB 2  OE 2  EB 2
x
10  X  7
2
2
10 cm
2
C
20cm
10 cm
O
D
X 2  100  49  51
X  51  7.1 cm
Lesson 8-4: Arcs and Chords
9