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Objectives:
To use the relationship between a
radius and a tangent
To use the relationship between two
tangents from one point


Tangent to a Circle  a line in the plane of
the circle that intersects the circle in exactly
one point.
Point of Tangency  the point where a circle
and a tangent intersect.
A
Point of Tangency
B
Tangent to a Circle

Theorem 12.1
◦ If a line is tangent to a circle, then the line is
perpendicular to the radius drawn to the point of
tangency.
A
O
AB ⊥ OP
P
B

Theorem 12.2
◦ If a line in the plane of a circle is perpendicular to a
radius at its endpoint on the circle, then the line is
tangent to the circle.
A
O
P
B
AB ⊥ ΘO

Theorem 12.3
◦ The two segments tangent to a circle from a point
outside the circle are congruent.
A
B
O
~
AB = CB
C
Homework
# 32
Due Tuesday (April 30)
Page 665 – 666
◦# 1 – 20 all

Objectives:
To use congruent chords, arcs, and central
angles
To recognize properties of lines through
the center of a circle

Chord  a segment whose endpoints are on
a circle
P
PQ and PQ
O
Q

Theorem 12.4
◦ Within a circle or in congruent circles:
1. Congruent central angles have congruent
chords
2. Congruent chords have congruent arcs
3. Congruent arcs have congruent central angles

Theorem 12.5
◦ Within a circle or in congruent circles:
1. Chords equidistant from the center are
congruent
2. Congruent chords are equidistant from the
center

Theorem 12.6
◦ In a circle, a diameter that is perpendicular to a
chord bisects the chord and its arcs.

Theorem 12.7
◦ In a circle, a diameter that bisects a chord (that is
not a diameter) is perpendicular to the chord.

Theorem 12.8
◦ In a circle, the perpendicular bisector of a chord
contains the center of the circle.
 Homework
#33
 Due Wed/Thurs (May 1/2)
 Page 673 – 674
 # 1 – 19 all

Objectives:
To find the measure of an inscribed angle
To find the measure of an angle formed
by a tangent and a chord


Inscribed Angle  angle in a circle in which the
vertex is on the circle and the sides of the angle
are chords of the circle
Intercepted Arc  arc created by drawing an
inscribed angle
A
Intercepted
Arc
B
Inscribed
Angle
C

Theorem 12.9 – Inscribed Angle Theorem
◦ The measure of an inscribed angle is half the
measure of its intercepted arc.
A
m<B =
B
C
1
mAC
2

Corollaries to the Inscribed Angle Theorem
◦ 1. Two inscribed angles that intercept the same arc
are congruent.
◦ 2. An angle inscribed in a semicircle is a right angle
◦ 3. The opposite angles of a quadrilateral inscribed
in a circle are supplementary.

Theorem 12.10
◦ The measure of an angle formed by a tangent and a
chord is half the measure of the intercepted arc.
m<C =
B
1
mBDC
2
B
D
D
C
C
 Homework
#34
 Due Friday (May 03)
 Page 681 – 682
◦# 1 – 22 all

Objectives:
To find the measures of angles formed by
chords, secants, and tangents
To find the lengths of segments
associated with circles

Secant  a line that intersects a circle at two
points.
A
O
B

Theorem 12.11
◦ The measure of an angle formed by two lines that
intersect inside a circle is half the sum of the
measures of the intercepted arcs.
x°
1
y°
m<1 =
1
(x
2
+ y)

Theorem 12.11 cont.
◦ The measure of an angle formed by two lines that
intersect outside a circle is half the difference of the
measures of the intercepted arcs.
y°
x°
1
m<1 =
1
(x
2
– y)

Theorem 12.12
◦ For a given point and circle, the product of the
lengths of the two segments from the point to the
circle is constant along any line through the point
and circle.
a
x
c
w
P
y
b
d
z
(w + x)w = (y + z)y
a·b=c·d
t
y
z
(y + z)y = 𝑡 2
Homework
#35
Due Friday (May 03)
Page 691
◦# 1 – 14 all