Download Geometry 10-1 Circles and Circumference A. Parts of Circles 1. A

Geometry 10-1 Circles and Circumference
A. Parts of Circles
1. A circle is the locus of all points equidistant from a given point called
the center of the circle.
2. A circle is usually named by its ________ point.
3. The circle below is called circle K, written d K .
A C
Any segment with endpoints
that are on the circle is a
chord of the circle. AB and
CD are chords.
Any segment with endpoints that
are the _______ and a point on
the circle is a radius. KF , KC ,
and KD are radii of the circle.
K
B
A chord that passes
through the ________ of
the circle is a diameter
of the circle. CD is a
diameter of the circle.
F
D
Ex 1: Identify parts of the circle.
a.) Name the circle.
B
A
E
b.) Name a radius of the circle.
C
c.) Name a chord of the circle.
D
d.) Name a diameter of the circle.
4. The distance from the center of the circle to any point on the circle is
always the same, therefore, all radii are congruent.
5. A diameter is composed of two radii, so,
d = 2r
P
Ex 2: Circle R has diameters ST and QM .
a.) If ST = 18, find RS.
b.) If RM = 24, find QM.
c.) If RN = 2, find RP.
S
Q
R
N M
T
Ex 3: The diameters of d X , d Y , and d Z are 22 millimeters, 16
millimeters, and 10 millimeters respectively.
a.) Find EZ.
E
X
Y
F
Z
G
b.) Find XF.
B. Circumference.
1. The circumference is the distance around a circle.
2. The circumference around a circle is often represented by the letter C.
3. The ratio between the circumference of a circle and it’s diameter is
about 3:1. For each diameter, circumference is a little more than 3.
4. For a circumference of C units, diameter of d, or radius of r,
C = π d or C = 2rπ
Ex 4: Find the missing value for each circle.
a.) Find C if r = 13 inches.
b.) Find C if d = 6 millimeters.
c.) Find d and r to the nearest
Ex 5: Find the exact circumference for d K .
3 2
K
HW: Geometry 10-1 p. 526-528
17-55 odd, 58-60, 63-64, 75-80
Hon: 61, 65, 73-74
Geometry 10-2 Angles and Arcs
A. Angles and Arcs.
1. A central angle has the __________ of the circle as the vertex and its
sides contain two radii of the circle.
2. Sum of Central Angles - the sum of the
measures of a circle with no interior points in
1 3
common is 360.
2
m∠1 + m∠2 + m∠3 = 360
R
Ex 1: Refer to d T .
a.) Find m∠RTS .
Q
S
(8 x − 4)o
T
(13x − 3)o
20xo
b.) Find m∠QTR .
(5 x + 5)o
U
V
example
minor arc
A
B
major arc
»AC
110
o
C
named:
by the two letters of
the endpoints »AC
arc degree
measure =
the measure of the
central angle is less
than 180.
m∠ABC = 110o
so m »AC = 110
D
E 40
o
semicircle
¼
DFE
G
F
J
¼
JML
M
K
¼
JKL
L
by the letters of two
by the letters of the
endpoints and another two endpoints and
¼
another point on the
point on the arc DFE
¼ and JML
¼
arc JKL
¼ = ____
greater than 180, but
mJKL
less than 360.
¼ = ____
mJML
¼ = 360 − mDE
»
mDFE
¼ = ___
mDFE
3. Theorem 10-1 - In the same or in congruent circles, two arcs are
congruent if and only if their corresponding central angles are congruent.
P
4. Postulate 10-1 Arc Addition Postulate - The
measure of an arc formed by two adjacent arcs is
the sum of the measures of the two arcs.
» + mRQ
» = mPQR
¼ .
In d S , mPQ
S
Q
R
Ex 3: In d P , m∠NPM = 46 , PL bisects ∠KPM , and OP ⊥ KN . Find
L
each measure.
K
»
a.) mOK
¼
b.) mLM
M
P
J
¼
c.) mJKO
N
O
Ex 4: This graph shows the percent of each type
of bicycle sold in the United States in 2001.
a.) Find the measurement of the central
angle representing each category.
Youth - 26%(360) = _____
Mountain - ____________ = ____
Comfort Road Hybrid -
Bicycles Bought In 2001 (by type)
Hybrid
9%
b.) Is the arc for the wedge named Youth
congruent to the arc for the combined
wedges named Road and Comfort?
B. Arc Length
1. An arc is part of a circle, so arc length is part of the
circumference.
Ex 5: In d B , AC = 9, and m∠ABD = 40 , Find the
length of »AD
HW Geometry 10-2 p. 533-535
15-43 odd, 47-51, 53, 57-62, 66, 69-76
Hon: 63, 67
Mountain
37%
Youth
26%
Comfort
21%
Road
7%
A
B
D
C
Geometry 10-3 Arcs and Chords
A. Arcs and Chords
1. Theorem 10-2 - In a circle or in congruent circles, two minor arcs are
congruent if and only if their corresponding chords are congruent.
Abb.
- In d , 2 minor arcs are ≅ , corr. chords are ≅ .
- In d , 2 chords are ≅ , corr. minor arcs are ≅ .
A
D
» , and if »AB ≅ CD
» , AB ≅ CD
Ex: If AB ≅ CD, »AB ≅ CD
Ex 1: Finish the proof of Theorem 10-2
C
U
V
» ≅ YW
»
Given: d X , UV
X
Prove: UV ≅ YW
Statements
» ≅ YW
»
1. d X , UV
2. ∠UXV ≅ ∠WXY
B
W
Reasons
Y
1. Given
2. If arcs are ____________________
_______________________________
3. UX ≅ VX ≅ WX ≅ YX
3.______________________________
4. __________________
4.____________
5. UV ≅ YW
5.________________
B. Diameters and Chords
1. Theorem 10-3 In a circle, if a diameter (or radius)
is perpendicular to a chord, then it bisects the chord
and its arc.
Ex: If BA ⊥ TV , then _______ and » ≅ » .
B
T
C
U
V
A
M
Ex 2: Circle W has a radius of 10 cm. Radius WL is
perpendicular to chord HK , which is 16 cm. long.
H
W
J
L
K
» = 53 , find mMK
¼ .
a.) If mHL
b.) Find JL.
2. Theorem 10-4 In a circle or in congruent circles, two chords are
congruent if and only if they are equidistant from the center.
Ex 3: Chords EF and GH are equidistant
from the center. If the radius of d P is 15 and
EF = 24 , find PR and RH.
Q
E
P
G
HW: Geometry 540-543
11-35 odd, 39-41, 48-49, 52-65
Hon: 36, 42-43
F
H
R
Geometry 10-4 Inscribed Angles
A. Inscribed angles
1. An inscribed angle is an angle that has its vertex and its sides contained
in the chords of the circle.
B
Vertex B is on the circle
»AC is the arc
intercepted by
∠ABC
AB and BC are
chords on the circle
A
C
2. Theorem 10-5 - Inscribed Angle Theorem - If an angle is inscribed in
a circle, then the measure of the angle equals one-half the measure of its
intercepted arc (or the measure of the intercepted
B
arc is twice the measure of the inscribed angle).
1 »
»
Ex: m∠ABC = (m AC
) or 2(m∠ABC ) = m AC
2
A
C
¼ = 20 , m »XY = 40 , mUZ
» = 108 , and mUW
¼ = mYZ
» .
Ex 1: In d F , mWX
W
X
Find the measures of the numbered angles.
Y
3
4
5
F
2
U
1
Z
T
3. Theorem 10-6 - If two inscribed angles of a circle (or congruent circles)
intercept congruent arcs or the same arc, then the angles are congruent.
B
A
Abbreviations:
Inscribed ∠ ’s of ≅ arcs are ≅ .
Inscribed ∠ ’s of same arc are ≅ .
∠DAC ≅ ∠ ____
D
C
Q
P
K
Ex 2:
Given: d C with QR ≅ GF ,
C
and JK ≅ HG
Prove: VPJK ≅VEHG
Reasons
1.
»
3. ∠GEF intercepts FG
»
∠QPR intercepts QR
4.
5.
6. VPJK ≅VEHG
3.
R
F
H
E
Statements
1. QR ≅ GF , JK ≅ HG
» ≅ GF
»
2. QR
J
G
2.
4.
5.
6.
B. Angles of Inscribed Polygons
1. Theorem 10-7 - If an inscribed angle intercepts a
semicircle, the angle is a right angle.
A
D
¼
ADC is a semicircle so m∠ABC = 90o
C
B
Ex 3: Triangles TVU and TSU are inscribed in
» ≅ SU
» . Find the measure of each
d P with VU
numbered angle if m∠2 = x + 9 and
m∠4 = 2 x + 6 .
V
3
1
T
Ex 4: Quadrilateral QRST is inscribed in d M .
If m∠Q = 87 and m∠R = 102 . Find m∠S and m∠T .
2. Theorem 10-8 - If a quadrilateral is inscribed in a
circle, then its opposite angles are supplementary.
HW: Geometry 10-4 p. 549-551
8-10, 13-16, 18-20, 22-29, 40-43, 46-58
Hon: 11, 17, 31-32, 35
U
4
2
S
Geometry 10-5 Tangents
A. Tangents
1. BC is tangent to d A , because the line
uuu
r
containing BC intersects the circle in exactly
one point.
2. This point is called the point of tangency.
A
B
C
3. Theorem 10-9 - If a line is tangent to a
circle, then it is perpendicular to the
radius drawn to the point of tangency.
K
suu
r
If AB is a tangent, AB ⊥ AK
A
B
S
Ex 1: RS is tangent to d Q at point R. Find the diameter..
20
P
16
R
Q
4. Theorem 10-10 - In a plane, if a line is ______________ to a radius of a
circle at the endpoint on the circle, then the line is a tangent to the circle.
Ex 2: a.) Determine whether
BC is tangent to d A .
b.) Determine whether
WE is tangent to d D .
E
16
A
7
D
7
10
7
B
9
10
C
5. Theorem 10-11 - If two segments from
the same exterior point are tangent to a
circle, they are congruent.
24
W
A
B
AB ≅ BC
C
Ex 3: Find x, assume that segments that
appear to be tangent are.
T
H
G
y-5
x+4
S
E
F
10
y
B. Circumscribed Polygons
1. Polygons can be circumscribed about a circle, or the circle is inscribed
in the polygon. (every side of the polygon is tangent to the circle.)
Ex 4: Find the perimeter of the triangle.
Given: AD = CB
D
C
6
E
19
A
HW: Geometry 10-5 p. 556-558
8-19, 23-25, 29-30, 33-40, 42-45
Hon: 20, 26
F
B
Geometry 10-6 Secants, Tangents, and Angle Measures
A. Intersections on or inside a circle.
1. A line that intersects a circle in exactly two points is called a
______________.
-A secant of a circle contains a chord of the circle.
2. Theorem 10-12 - If two secants intersect in the
interior of a circle, then the measure of an angle
formed is one-half the sum of the measures of the
arcs intercepted by the angle and its vertical angle.
» )
(m »AC + mDB
Examples: m∠1 =
2
m∠2 =
A
1
B
C
» )
(m »AD + mBC
2
F
» = 88 and mEH
¼ = 76
Ex 1: Find m∠4 if mFG
88
G
4
3
E
76
3. Theorem 10-13 - If a secant and a tangent
intersect at the point of tangency, then the
measure of each angle formed is ____________
the measure of its intercepted arc.
D
2
H
180o
R
» = 114o and mTS
º = 136o
Ex 2: Find m∠RPS if mPT
P
Q
S
114o
T
136o
B. Intersections outside of a circle.
1. Theorem 10 -14 - If two secants, a secant and a tangent, or two tangents
intersect in the exterior of a circle, then the measure of the angle formed is
one-half the positive difference of the measures of the intercepted arcs.
Two secants
Secant - Tangent
Two Tangents
D
D
E
D
B
B
B
C
C
A
A
C
A
1 »
1 »
» )
» )
− mBC
− mBC
m∠A = (mDE
m∠A = (mDC
2
2
1 ¼
» )
− mBC
m∠A = (mBDC
2
Ex 3: Find x.
141o
xo 62o
Ex
4: A jeweler wants to craft a
pendant with the shape shown.
Use the figure to determine
the measure of the arc at the
bottom of the pendant.
Ex 5: Find x.
55o
40o
HW: Geometry 10-6 p. 564-568
13-31odd, 34-39, 44, 47-55, 57-59
Hon: 28, 32, 40a
6x o
40o
Geometry 10-7 Special Segments in a Circle
A. Segments intersecting in a circle.
1. Theorem 10-15 - If two chords intersect in a circle,
then the products of the measures of the segments of the
chords are equal.
A
E
So AE ⋅ EC = DE ⋅ EB .
Ex 1: Find x.
12
8
B
C
D
x
9
Ex 2: Biologists often examine organisms under
microscopes. The circle represents the field of view
under the microscope with a diameter of 2 mm.
Determine the length of the organism is it is located
0.25 mm from the bottom of the field of view.
Round to the nearest hundredth of a millimeter.
0.25 mm
B. Segments intersecting outside a Circle
1. Theorem 10-16 - If two secant segments are drawn to a circle from an
___________ _________, the product of the measures of one secant
segment and its external secant segment is equal the product of the
measures of the other secant segment
A
B
and its external secant segment.
C
D
AB ⋅ AC = AD ⋅ AE
Ex 3: Find HJ if EF = 10, EH = 8, and FG = 24.
E
E
H
J
F
G
2. If a _______________ segment and a _______________ segment are
drawn to a circle from an exterior point, then the square of the measure
of the tangent segment is equal to the product of the measures of the
secant segment and its external
X
W
secant segment.
(WX ) 2 = WZ ⋅ WY
Y
Z
Ex 4: Find x, assume that segments that appear
to be tangent are.
6
3
Ex 5: Find x, assume that segments that appear to be tangent are.
x+2
x
x+4
HW: Geometry 10-7 p. 572-574
8-19, 23, 25-29, 34-41, 43-48
x
Geometry 10-8 Equations of Circles
A. Equation of a Circle
1. Suppose the center of a circle is at
(3, 2), and the radius is 4.
2. Use the distance formula to
determine the distance to a point on the
circle.
P(x, y)
(3, 2)
d = ( x2 − x1 ) 2 + ( y2 − y1 ) 2
4 = ( x − __) 2 + ( y − __) 2
16 = (_____) 2 + (_____) 2
this is the general form for an equation
of a circle.
3. An equation for a circle with center at (h, k) and radius of r units, is
( x − h) 2 + ( y − k ) 2 = r 2
Ex 1: Write an equation for each circle.
a.) center at (4, -3), r = 6
b.) center at (-12, -1), d = 16.
Ex 2: A circle with a diameter of 10 has its
center in the first quadrant. The lines
y = −3 and x = −1 are tangent to the circle.
Write an equation of the circle.
Ex 3: a.) Graph ( x − 2) 2 + ( y + 3)2 = 4
c.) Graph ( x − 3) 2 + y 2 = 16
HW: Geometry 10-8 p. 578-580
11-33odd, 35-37, 47-53
Hon: 38-39, 54-55