Post on 06-Sep-2018
Tangents to Circles
Section 12.1
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
diameter
radiusP
center
name of circle: P
Definition
Chord – a segment whose endpoints are points on the circle.
AB is a chord
B
A
Definition
Secant – a line that intersects a circle in two points.
MN is a secant
N
M
Definition
Tangent – a line in the plane of a circle that intersects the circle in exactly one point.
ST is a tangent
S
T
Example 1
Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius.
FCB
G
A
H
D
E
Id. CE
c. DF
b. EI
a. AH tangent
diameter
chord
radius
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 tangentcommon internal tangent
Example 2
Tell whether the common tangents are internal or external.
a. b.
common internal tangents 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
Q
P
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
Q
P
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
CD B
Aminor arc: AB
major arc: ABD
semicircle: BAD
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
B
mABC = mAB + mBC
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
CAB ≅ BC if and only if AB ≅ BC
Perpendicular Diameter Theorem If a diameter of a circle is perpendicular to a chord,
then the diameter bisects the chord and its arc.
D
F
GE
DE ≅ EF, DG ≅ FG
Perpendicular Diameter Converse If one chord is a perpendicular bisector of another
chord, then the first chord is a diameter.
L
MJ
K
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.
G
F
E
C
D
B
A
AB ≅ CD if and only if EF≅ EG.
Example 3
Tell whether CE is tangent to D.
45
43
11
D
E
C
Use the converse of the Pythagorean Theorem to see if the triangle is right.
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.
SP
R
T
If SR and ST are tangent to P, then SR ≅ ST.
Example 4
AB is tangent to C at B.AD is tangent to C at D.
Find the value of x.
11
x2 + 2
AC
D
BAD = AB
x2 + 2 = 11
x2 = 9
x = ±3
Example 1 Find the measure of each arc.
70°PN L
M
a. LM
c. LMN
b. MNL
70°
360° - 70° = 290°
180°
Example 2 Find the measures of the red arcs. Are the arcs congruent?
41°
41°
AC
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?
81°
C
A
D
E
mDE = mAC = 81°However, since the arcs are not of the same circle orcongruent circles, they are NOT congruent!
Example 4
A
(2x + 48)°(3x + 11)°
D
B
C
3x + 11 = 2x + 48
Find mBC.
x = 37
mBC = 2(37) + 48
mBC = 122°
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
inscribed angle
intercepted arc
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.
C
A
D Bm∠ADB =
12
mAB
Example 1
Find the measure of the blue arc or angle.
RS
QT
a.
mQTS = 2(90°) = 180°
b.80°
E
F G
m∠EFG = 12
(80°) = 40°
Congruent Inscribed Angles Theorem If two inscribed angles of a circle intercept
the same arc, then the angles are congruent.
A
CB
D
∠C ≅ ∠D
Example 2
It is given that m∠E = 75°. What is m∠F?
D
E
HF
Since ∠E and ∠F both interceptthe same arc, we know that theangles must be congruent.
m∠F = 75°
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
C
B∠B is a right angle if and only if ACis a diameter of the circle.
Inscribed Quadrilateral Theorem
A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.
C
E
F
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.
2x°
Q
A
B
C
a.
2x° = 90°
x = 45
b. z°y°
80°
120°
D
E
F
G
m∠D + m∠F = 180°
z + 80 = 180
z = 100
m∠G + m∠E = 180°
y + 120 = 180y = 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.
2 1
B
A
Cm∠1 = 12
mAB
m∠2 = 12
mBCA
Example 1m
102°
T
R
S
Line m is tangent to the circle. Find mRST
mRST = 2(102°)
mRST = 204°
Try This!Line m is tangent to the circle. Find m∠1
m
150°
1
T
Rm∠1 =
12
(150°)
m∠1 = 75°
Example 2
(9x+20)°
5x°
D
B
CA
BC is tangent to the circle. Find m∠CBD.
2(5x) = 9x + 20
10x = 9x + 20
x = 20
m∠CBD = 5(20°)
m∠CBD = 100°
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 = 12
(mCD + mAB)
m∠2 = 12
(mAD + mBC)2
1
A
C
D
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
1
BA
C
m∠1 = 12
(mBC - mAC)
2
P
RQ
m∠2 = 12
(mPQR - mPR)
3
XW
YZ
m∠3 = 12
(mXY - mWZ)
Example 3 Find the value of x.
174°
106°
x°
P
R
Q
S
x° = 12
(mPS + mRQ)
x° = 12
(106°+174°)
x = 12
(280)
x = 140
Try This! Find the value of x.
120°
40°
x°
T
R
S
Ux° =
12
(mST + mRU)
x° = 12
(40°+120°)
x = 12
(160)
x = 80
Example 4 Find the value of x.
200°
x° 72°
72° = 12
(200° - x°)
144 = 200 - x
x = 56
Example 5 Find the value of x.
mABC = 360° - 92°
mABC = 268° x°92°
C
AB
x = 12
(268 - 92)
x = 12
(176)
x = 88
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 chord.
E
C
D
A
B
EA ⋅ EB = EC ⋅ ED
Example 1• Find the value of x.
x
96
3
E
B
D
A
C3(6) = 9x
18 = 9x
x = 2
Try This!• Find the value of x.
x 9
18
12E
B
D
A
C
9(12) = 18x
108 = 18x
x = 6
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.
C
A
B
ED
EA ⋅ EB = EC ⋅ ED
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.
C
A
E
D
(EA)2 = EC ⋅ ED
Example 2• Find the value of x.
LM ⋅ LN = LO ⋅ LP
9(20) = 10(10+x)
180 = 100 + 10x
80 = 10x
x = 8 x
10
11
9
O
M
N
L
P
Try This!• Find the value of x.
x
1012
11
H
GF
E
D
DE ⋅ DF = DG ⋅ DH
11(21) = 12(12 + x)
231 = 144 + 12x
87 = 12x
x = 7.25
Example 3• Find the value of x.
x
12
24
D
BC
A
CB2 = CD(CA)
242 = 12(12 + x)
576 = 144 + 12x
432 = 12x
x = 36
Try This!• Find the value of x.
3x5
10
Y
W
X Z
WX2 = XY(XZ)
102 = 5(5 + 3x)
100 = 25 + 15x
75 = 15x
x = 5