Transcript Tangents to Circles Section 10.1

Tangents to Circles
Section 10.1
Essential Questions


How do I identify segments and lines related to
circles?
How do I use properties of a tangent to a
circle?
Definitions




A circle is the set of all points in a plane that are
equidistant from a given point called the center of the
circle.
Radius – the distance from the center to a point on the
circle
Congruent circles – circles that have the same radius.
Diameter – the distance across the circle through its
center
Diagram of Important Terms
center
radius
P
diameter
name of circle:
P
Definition

Chord – a segment whose endpoints are points on the
circle.
B
A
AB is a chord
Definition

Secant – a line that intersects a circle in two points.
N
M
MN is a secant
Definition

Tangent – a line in the plane of a circle that intersects
the circle in exactly one point. T
S
ST is a tangent
Example 1

Tell whether the line or segment is best described as a chord, a
secant, a tangent, a diameter, or a radius.
H
a. AH
tangent
b. EI
diameter
B
C
c. DF
d. CE
E
F
chord
G
I
radius
A
D
Definition

Tangent circles – coplanar circles that intersect
in one point
Definition

Concentric circles – coplanar circles that have
the same center.
Definitions

Common tangent – a line or segment that is tangent to two
coplanar circles
–
–
Common internal tangent – intersects the segment that joins the
centers of the two circles
Common external tangent – does not intersect the segment that joins
the centers of the two circles
common external tangent
common internal tangent
Example 2

Tell whether the common tangents are internal or external.
a.
common internal tangents
b.
common external tangents
More definitions


Interior of a circle – consists of the points that
are inside the circle
Exterior of a circle – consists of the points that
are outside the circle
Definition

Point of tangency – the point at which a tangent line
intersects the circle to which it is tangent
point of tangency
Perpendicular Tangent Theorem

If a line is tangent to a circle, then it is perpendicular to
the radius drawn to the point of tangency.
l
P
Q
If l is tangent to
Q at P, then l  QP.
Perpendicular Tangent Converse

In a plane, if a line is perpendicular to a radius of a
circle at its endpoint on the circle, then the line is
tangent to the circle.
l
P
Q
If l  QP at P, then l is tangent to
Q.
Definition
 Central angle – an angle whose vertex is the center of
a circle.
central angle
Definitions
 Minor arc – Part of a circle that measures less than
180°
 Major arc – Part of a circle that measures between
180° and 360°.
 Semicircle – An arc whose endpoints are the
endpoints of a diameter of the circle.
Note : major arcs and semicircles are named with three
points and minor arcs are named with two points
Diagram of Arcs
A
minor arc: AB
major arc: ABD
semicircle: BAD
D
C
B
Definitions
 Measure of a minor arc – the measure of its central
angle
 Measure of a major arc – the difference between 360°
and the measure of its associated minor arc.
Arc Addition Postulate
 The measure of an arc formed by two adjacent arcs is
the sum of the measures of the two arcs.
A
C
mABC = mAB + mBC
B
Definition
 Congruent arcs – two arcs of the same
circle or of congruent circles that have the
same measure
Arcs and Chords Theorem
 In the same circle, or in congruent circles, two minor
arcs are congruent if and only if their corresponding
chords are congruent.
A
B
AB  BC if and only if AB  BC
C
Perpendicular Diameter
Theorem
 If a diameter of a circle is perpendicular to a chord,
then the diameter bisects the chord and its arc.
F
DE  EF, DG  FG
E
G
D
Perpendicular Diameter
Converse
 If one chord is a perpendicular bisector of another
chord, then the first chord is a diameter.
J
M
K
L
JK is a diameter of the circle.
Congruent Chords Theorem
 In the same circle, or in congruent circles, two chords
are congruent if and only if they are equidistant from
the center.
C
G
E
AB  CD if and only if EF EG.
D
B
F
A
Example 3
Tell whether CE is tangent to
D.
Use the converse of the Pythagorean
Theorem to see if the triangle is right.
C
43
E
45
11
D
112
+
432
?
452
121 + 1849 ? 2025
1970  2025
CED is not right, so CE is not tangent to
D.
Congruent Tangent Segments Theorem

If two segments from the same exterior point are
tangent to a circle, then they are congruent.
R
P
S
T
If SR and ST are tangent to
P, then SR  ST.
Example 4
AB is tangent to
AD is tangent to
Find the value of x.
AD = AB
x2 + 2 = 11
x2 = 9
x = 3
D
C at B.
C at D.
x2 + 2
A
C
11
B
Example 1

Find the measure of each arc.
a. LM
70°
N
b. MNL
360° - 70° = 290°
c. LMN
180°
P
L
70
M
Example 2

Find the measures of the red arcs. Are the arcs congruent?
A
C
41
41
D
E
mAC = mDE = 41
Since the arcs are in the same circle, they are congruent!
Example 3

Find the measures of the red arcs. Are the arcs congruent?
A
D
81
E
C
mDE = mAC = 81
However, since the arcs are not of the same circle or
congruent circles, they are NOT congruent!
Example 4
B
Find mBC.
(3x + 11)
(2x + 48)
3x + 11 = 2x + 48
A
x = 37
D
mBC = 2(37) + 48
mBC = 122
C
Definitions
 Inscribed angle – an angle whose vertex is on a circle
and whose sides contain chords of the circle
 Intercepted arc – the arc that lies in the interior of an
inscribed angle and has endpoints on the angle
intercepted arc
inscribed angle
Measure of an Inscribed Angle Theorem
 If an angle is inscribed in a circle, then its measure is
half the measure of its intercepted arc.
A
mADB =
1
2
C
mAB
D
B
Example 1
 Find the measure of the blue arc or angle.
a.
S
R
E
b.
80
F
T
G
Q
mQTS = 2(90) = 180
mEFG =
1
2
(80) = 40
Congruent Inscribed Angles Theorem
If two inscribed angles of a circle intercept
the same arc, then the angles are
congruent.
A
B
C
D
C  D
Example 2
It is given that mE = 75. What is mF?
D
Since E and F both intercept
the same arc, we know that the
angles must be congruent.
E
mF = 75
F
H
Definitions
 Inscribed polygon – a polygon whose vertices all lie on a
circle.
 Circumscribed circle – A circle with an inscribed polygon.
The polygon is an inscribed polygon and
the circle is a circumscribed circle.
Inscribed Right Triangle Theorem
 If a right triangle is inscribed in a circle, then the hypotenuse
is a diameter of the circle. Conversely, if one side of an
inscribed triangle is a diameter of the circle, then the triangle
is a right triangle and the angle opposite the diameter is the
right angle.
A
B is a right angle if and only if AC
is a diameter of the circle.
B
C
Inscribed Quadrilateral Theorem
 A quadrilateral can be inscribed in a circle if and only if
its opposite angles are supplementary.
E
F
C
D
G
D, E, F, and G lie on some circle, C if and only if
mD + mF = 180 and mE + mG = 180.
Example 3
 Find the value of each variable.
a.
B
Q
A
D
b.
G
y
F
C
x = 45
E
80
2x
2x = 90
z
120
mD + mF = 180
z + 80 = 180
z = 100
mG + mE = 180
y + 120 = 180
y = 60
Tangent-Chord Theorem

If a tangent and a chord intersect at a point on a circle, then
the measure of each angle formed is one half the measure of
its intercepted arc.
B
m1 =
m2 =
1
2
1
2
mAB
mBCA
C
2
1
A
Example 1
Line m is tangent to the circle. Find mRST
m
R
102
mRST = 2(102)
S
mRST = 204
T
Try This!
Line m is tangent to the circle. Find m1
m1 =
1
2
R
(150)
1
m
m1 = 75
150
T
Example 2
BC is tangent to the circle. Find mCBD.
C
A
(9x+20)
5x
2(5x) = 9x + 20
10x = 9x + 20
x = 20
D
mCBD = 5(20)
mCBD = 100
B
Interior Intersection Theorem

If two chords intersect in the interior of a circle, then the
measure of each angle is one half the sum of the measures of
the arcs intercepted by the angle and its vertical angle.
m1 =
m2 =
1
2
1
2
(mCD + mAB)
D
A
2
(mAD + mBC)
1
C
B
Exterior Intersection Theorem

If a tangent and a secant, two tangents, or two
secants intersect in the exterior of a circle,
then the measure of the angle formed is one
half the difference of the measures of the
intercepted arcs.
Diagrams for Exterior
Intersection Theorem
B
A
P
1
2
C
m1 =
1
2
Q
R
X
(mBC - mAC)
m2 =
W
3
Z
m3 =
1
2
Y
(mXY - mWZ)
1
2
(mPQR - mPR)
Example 3

P
Find the value of x.
106
Q
x =
x =
x=
1
2
1
2
1
2
(mPS + mRQ)
x
S
(106+174)
174
(280)
x = 140
R
Try This!

Find the value of x.
x =
x =
1
2
1
2
x=
40
T
S
(mST + mRU)
x
U
(40+120)
1
2
R
(160)
x = 80
120
Example 4

Find the value of x.
72 =
1
2
(200 - x)
144 = 200 - x
x = 56
200
x 72
Example 5

Find the value of x.
A
mABC = 360 - 92
mABC = 268
x=
B
92
1
(268 - 92)
2
1
x = (176)
2
x = 88
C
x
Chord Product Theorem
• If two chords intersect in the interior of 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
C chord.
B
E
EA  EB = EC  ED
D
A
Example 1
• Find the value of x.
B
3
3(6) = 9x
18 = 9x
A
9
E
x
C
6
x= 2
D
Try This!
• Find the value of x.
B
C
9
x
9(12) = 18x
108 = 18x
E
12
A
18
x= 6
D
Secant-Secant Theorem
• If two secant segments share the same endpoint
outside a circle, then the product of the length of one
secant segment and the length of its external segment
equals the product of the length of the other secant
segment and the length of its external segment.
B
EA  EB = EC  ED
A
D
E
C
Secant-Tangent Theorem
• If a secant segment and a tangent segment share an
endpoint outside a circle, then the product of the
length of the secant segment and the length of its
external segment equals the square of the length of
the tangent segment.
A
(EA)2 = EC  ED
E
C
D
Example 2
• Find the value of x.
N
11
LM  LN = LO  LP
M
9
9(20) = 10(10+x)
180 = 100 + 10x
80 = 10x
x= 8
L
10
O
x
P
Try This!
D
• Find the value of x.
11
E
DE  DF = DG  DH
10
12
F
11(21) = 12(12 + x)
x
231 = 144 + 12x
87 = 12x
x = 7.25
H
G
Example 3
• Find the value of x.
C
CB2
= CD(CA)
242 = 12(12 + x)
576 = 144 + 12x
432 = 12x
x = 36
24
B
12
D
x
A
Try This!
• Find the value of x.
W
WX2 = XY(XZ)
102 = 5(5 + 3x)
100 = 25 + 15x
75 = 15x
x= 5
10
X
5
Y
3x
Z