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Tangents to Circles(with Circle Review)
How do I identify segments and lines related to circles?
How do I use properties of a tangent to a circle?
Essential Questions
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
Definitions
Diagram of Important Terms
diameter
radiusP
center
name of circle: P
Chord – a segment whose endpoints are points on the circle.
Definition
AB is a chord
B
A
Secant – a line that intersects a circle in two points.
Definition
MN is a secant
N
M
Tangent – a line in the plane of a circle that intersects the circle in exactly one point.
Definition
ST is a tangent
S
T
Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius.
Example 1
FC
B
G
A
H
D
E
Id. CE
c. DF
b. EI
a. AH tangent
diameter
chord
radius
Tangent circles – coplanar circles that intersect in one point
Definition
Concentric circles – coplanar circles that have the same center.
Definition
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
Definitions
common external tangentcommon internal tangent
Tell whether the common tangents are internal or external.
Example 2
a. b.
common internal tangents common external tangents
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
More definitions
Point of tangency – the point at which a tangent line intersects the circle to which it is tangent
Definition
point of tangency
If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
Perpendicular Tangent Theorem
l
Q
P
If l is tangent to Q at P, then l QP.
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.
Perpendicular Tangent Converse
l
Q
P
If l QP at P, then l is tangent to Q.
Central angle – an angle whose vertex is the center of a circle.
Definition
central angle
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
Definitions
Diagram of Arcs
CD B
Aminor arc: AB
major arc: ABD
semicircle: BAD
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.
Definitions
The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.
Arc Addition Postulate
A
C
B
mABC = mAB + mBC
Congruent arcs – two arcs of the same circle or of congruent circles that have the same measure
Definition
In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.
Arcs and Chords Theorem
A
B
C
AB BC if and only if AB BC
If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.
Perpendicular Diameter Theorem
D
F
G
EDE EF, DG FG
If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.
Perpendicular Diameter Converse
L
M
J
K
JK is a diameter of the circle.
In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.
Congruent Chords Theorem
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.
If two segments from the same exterior point are tangent to a circle, then they are congruent.
Congruent Tangent Segments Theorem
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
Find the measure of each arc.
Example 1
70PN L
M
a. LM
c. LMN
b. MNL
70°
360° - 70° = 290°
180°
Find the measures of the red arcs. Are the arcs congruent?
Example 2
41
41
AC
D
E
mAC = mDE = 41Since the arcs are in the same circle, they are congruent!
Find the measures of the red arcs. Are the arcs congruent?
Example 3
81
C
A
D
E
mDE = mAC = 81However, 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
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
Definitions
inscribed angle
intercepted arc
If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.
Measure of an Inscribed Angle Theorem
C
A
D BmADB =
1
2mAB
Find the measure of the blue arc or angle.
Example 1
RS
QT
a.
mQTS = 2(90 ) = 180
b.80
E
FG
mEFG = 1
2(80 ) = 40
If two inscribed angles of a circle intercept the same arc, then the angles are congruent.
Congruent Inscribed Angles Theorem
A
CB
D
C D
Example 2
It is given that mE = 75 . What is mF?
D
E
HF
Since E and F both interceptthe same arc, we know that theangles must be congruent.
mF = 75
Inscribed polygon – a polygon whose vertices all lie on a circle.
Circumscribed circle – A circle with an inscribed polygon.
Definitions
The polygon is an inscribed polygon and the circle is a circumscribed circle.
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.
Inscribed Right Triangle Theorem
A
C
BB is a right angle if and only if ACis a diameter of the circle.
A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.
Inscribed Quadrilateral Theorem
C
E
F
DG
D, E, F, and G lie on some circle, C if and only if mD + mF = 180 and mE + mG = 180 .
Find the value of each variable.
Example 3
2x
Q
A
B
C
a.
2x = 90
x = 45
b. z
y
80
120
D
E
F
G
mD + mF = 180
z + 80 = 180
z = 100
mG + mE = 180
y + 120 = 180
y = 60
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.
Tangent-Chord Theorem
21
B
A
Cm1 = 1
2mAB
m2 = 1
2mBCA
Example 1
m
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 m1
m
150
1
T
Rm1 =
1
2(150 )
m1 = 75
Example 2
(9x+20)
5x
D
B
CA
BC is tangent to the circle. Find mCBD.
2(5x) = 9x + 20
10x = 9x + 20
x = 20
mCBD = 5(20 )
mCBD = 100
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.
Interior Intersection Theorem
m1 = 1
2(mCD + mAB)
m2 = 1
2(mAD + mBC)
21
A
C
D
B
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.
Exterior Intersection Theorem
Diagrams for Exterior Intersection Theorem
1
BA
C
m1 = 1
2(mBC - mAC)
2
P
RQ
m2 = 1
2(mPQR - mPR)
3
X
W
YZ
m3 = 1
2(mXY - mWZ)
Find the value of x.
Example 3
174
106
x
P
R
Q
S
x = 1
2(mPS + mRQ)
x = 1
2(106+174 )
x = 1
2(280)
x = 140
Find the value of x.
Try This!
120
40
x
T
R
S
U
x = 1
2(mST + mRU)
x = 1
2(40+120 )
x = 1
2(160)
x = 80
Find the value of x.
Example 4
200
x 72
72 = 1
2(200 - x )
144 = 200 - x
x = 56
Find the value of x.
Example 5
mABC = 360 - 92
mABC = 268 x92
C
AB
x = 1
2(268 - 92)
x = 1
2(176)
x = 88
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.
Chord Product Theorem
E
C
D
A
B
EA EB = EC ED
Find the value of x.
Example 1
x
96
3
E
B
D
A
C3(6) = 9x
18 = 9x
x = 2
Find the value of x.
Try This!
x 9
18
12
E
B
D
A
C
9(12) = 18x
108 = 18x
x = 6
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.
Secant-Secant Theorem
C
A
B
ED
EA EB = EC ED
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.
Secant-Tangent Theorem
C
A
E
D
(EA)2 = EC ED
Find the value of x.
Example 2
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
Find the value of x.
Try This!
x
1012
11
H
GF
E
D
DE DF = DG DH
11(21) = 12(12 + x)
231 = 144 + 12x
87 = 12x
x = 7.25
Find the value of x.
Example 3
x
12
24
D
BC
A
CB2 = CD(CA)
242 = 12(12 + x)
576 = 144 + 12x
432 = 12x
x = 36
Find the value of x.
Try This!
3x5
10
Y
W
X Z
WX2 = XY(XZ)
102 = 5(5 + 3x)
100 = 25 + 15x
75 = 15x
x = 5
