CIRCLES
LINE SEGMENT RELATIONSHIPS IN A CIRCLE
THEOREM: A line tangent to a circle is perpendicular to the radius drawn to the point
of tangency.
O
THEOREM: If a line is perpendicular to a radius at a point P on the circle, then the line
is tangent to the circle at P.
THEOREM: Two tangent segments to a circle from the same exterior point have equal
lengths.
R
O
T
S
EXERCISE:
RS is tangent to circle O at L.
L
O
W
1. What is the measure of <SLO?
2. If OW = 3, find OL and LW.
3. If OW = 3, OR is drawn and OR = 5, find LR.
4. If OW = 3 and SL = 3, find OS.
THEOREM: In the same circle, or in equal circles, equal chords have equal arcs.
R
Equal circles O and C
If AB = RS, then mAB = mRS
O
B
C
S
A
THEOREM: In the same circle, or in equal circles, equal arcs have equal chords.
THEOREM: A line passing through the center of the circle and perpendicular to a
chord bisects the chord and its arcs.
C
Given: Circle O
Chord AB intersects diameter CD at E
CD  AB
Conclusions: AE = EB
mAD = mDB
mAC = mCB
O
B
D
A
EXERCISE:
B
In circle O, AB  CD.
C
O
A
D
1. CE = 3, OE = 4, OD = ?
2. OC = 17, OE = 8, CD = ?
3. CD = 24, OE = 5, OB = ?
THEOREM: If two chords intersect within a circle, then the product of the lengths of
the segments of one chord is equal to the product of the lengths of the
segments of the other chord.
D
A
E
B
(AE)(EB) = (CE)(ED)
O
C
EXERCISES:
1.Find the measure indicated by ?.
R
L
S
M
a. KS = 5, KL = 3, KM = 10, RK = ?
O
b. KM = 7, RK = 14, KS = 2, KL = ?
c. RK = 30, KL = 20, ML = 80, RS = ?
D
2. Find the lengths of all four chord segments.
A
E
a. ED = r, CE = r + 5, AE = 2r, EB = r + 1
B
C
b. ED = r  3, CE = 2r, AE = r, AB = 15
THEOREM: Let two secants be drawn to a circle from an outside point.
Then the product of the lengths of one secant and its external segment
equals the product of the lengths of the other secant and its external
segment.
(AC)(AB) = (AE)(AD)
B
A
C
D
E
THEOREM: Let a tangent and a secant be drawn to a circle from an outside point.
Then the square of the length of the tangent segment is equal to the
product of the lengths of the secant and its external segment.
(WX)2 = (WZ)(WY)
X
W
Y
Z
EXERCISES:
In each of the following, find the length indicated by y or z.
1.
2.
2.
4.