8-3 & 8-4 TANGENTS, ARCS & CHORDS 5 Theorems. THEOREM 1: a line that intersects a circle is tangent...
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8-3 & 8-4 TANGENTS, ARCS & CHORDS 5 Theorems
Transcript of 8-3 & 8-4 TANGENTS, ARCS & CHORDS 5 Theorems. THEOREM 1: a line that intersects a circle is tangent...
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.tangent
tangent
EXAMPLES: SOLVE.
FIND DC
2.4 cm
1.8 cm
7.0 cm
E
A
C
F
B
D
FIND CE and EA
(Segments that appear to be tangent are.)
Lesson 8-4: Arcs and Chords 5
Theorem 3:
If AB CD then AB CD
127 , .Given mAB find the mCD
In a circle, two chords are congruent iff their corresponding minor arcs are congruent.
E
A B
CD
Example:
127
Since mAB mCD
mCD
Lesson 8-4: Arcs and Chords 6
Theorem 4:
sec .
If DC AB then DC bi ts AB and AB
AE BE and AC BC
120 , .If mAB find mAC
In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects the chord and its arc.
E
D
A
C
BExample: If AB = 5 cm, find AE.
52.5
2 2
120, 60
2 2
ABAE AE cm
mABmAC mAC
Lesson 8-4: Arcs and Chords 7
Theorem 5:
In a circle, two chords are congruent if and only if they are equidistant from the center.
O
A B
C
DF
EExample: If AB = 5 cm, find CD.
Since AB = CD, CD = 5 cm.
CD AB iff OF OE
Lesson 8-4: Arcs and Chords 8
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?
secOD bi ts AB
8cm
15cm
O
A BD
∆ODB is a right triangle and
2 2 2
2 2 2
AB 15DB= = =7.5 cm
2 2OD=8 cmOB =OD +DBOB =8 +(7.5) =64+56.25=120.25OB= 120.25 11 cm
Solution:
x
Lesson 8-4: Arcs and Chords 9
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.
14sec . 7
2 2
ABOE bi ts AB EB cm
10 cm10 cm
20cm O
A B
DC
14 cm
xE
Solution:
OB (radius) = 10 cm∆EOB is a right triangle.2 2 2
2 2 2
2
10 7
100 49 51
51
OB OE EB
X
X
X
7.1 cm
