Unit 6: Circles Fall 2014 10.6 Circles and Arcs
Chapter 10 Notes
definitions
a) circle:____________________________________
●
O
congruent circles have _____________________.
All circles are ______________________.
b) concentric circles
c) diameter _______________________________________________________
d) central angle _________________________________
e) arc ________________________________________
•
semicircle _____________________________________
•
minor arc _______________________________________
•
major arc ______________________________________
•
adjacent arcs ____________________________________
1) Label the following circle as instructed by your teacher.
Name ….
a) 3 minor arcs _____________________
b) 3 major arcs ______________________
c) 2 semicircles ____________________
d) 2 adjacent arcs ___________________
II) Arc Addition Postulate
2) Find the measure of the following angles and arcs of circle O.
M
q = ______
a ) mDX
q = ______
b) mYXW
c) m∠WOX = ______
q = ______
d ) mMDW
e) m∠MOW = ______
p = ______
f )mYX
Y
40˚
D
O
O
56˚
W
X
1
Central Angles
Unit 6: Circles Fall 2014 Circumference: _____________________________
Arc Length ___________________________
Example:
Find the circumference of the following circle with radius 6 ft.
EXACT: _____________ Approximate: __________
p
AB ≅ BD
Find the exact LENGTHS of the following arcs given p
q = ________
a) m p
AD = _________ b) m p
AE = ________ c) m ADF
p
p
q
AD = _________
AE = _________
ADF = ________
Congruent Arcs
Draw and label the indicated arc and then find a) the measure of the arc and b) the length the arc. Be careful to
use appropriate notation and appropriate units!
3) : A with radius = 10 cm
p with central angle: 60°
PQ
4) : B with radius = 3 m
p
XY with central angle: 120°
5) A circular swimming pool 16 feet in diameter will be enclosed in a circular fence 4 ft from the pool. What is
the length of the fencing material needed? Round your answer to the nearest whole number.
Algebra Review: Simplify each rational expression.
5 x 3 yz 2
1.
20 xy 5
24a 2b 6
2.
36a 4b
( 3m n )
3.
(15m )( 2m )
2
4
2
5
4.
9 x 2 + 18
27
Unit 6: Circles Fall 2014 10.7 Area of Circles, Sectors , and Segments
Chapter 10 Notes
Sector: ________________________________
Area of a sector
Area of a circle
Find the area of the following sectors. Round your answer to the nearest whole number.
1)
2) A circle has a diameter of 10 cm.
3) How much more pizza is in a 14-in.What is the area of a sector bounded
by a 208° major arc?
diameter pizza than a 12-in. pizza?
Segment:
AREA OF SEGMENT = (AREA OF __________) - (AREA OF ___________)
Find the area of the shaded segment. Leave your answers in EXACT form.
6)
7)
8)
Algebra Review: Simplify each rational expression. You can cancel factors, but not terms!
1.
2a 3 − 8a
4a 2
2.
x2 + 7 x − 8
x2 − x
3.
3
16 − p 2
p2 + 3 p − 4
4.
x2 − 4
x2 + 4x + 4
Unit 6: Circles Fall 2014 12.1 Tangent Lines
tangent ____________________________________________
point of tangency ___________________________________
Theorem 12.1
Converse:
1. Given TJ is tangent to circle O at T.
a) If TJ = 12 and TO= 9,
b) If m∠TOJ = 60D and TO=10,
c) If TO = 5 and JO = 8,
then KJ = _________
then KJ = _________
then TJ = ___________
2) If x = 117°,
find m∠CBA
3) DE is tangent at point D,
find x.
THM 12-3
Two segments that are ______________ to the same circle from a
point outside the circle, must be _________
A
O
4) A belt fits tightly around two pulleys as shown.
Find the distance between the centers of the pulleys.
Given: AB and BC are tangent to : O
Prove: AB = BC
B
C
4
Unit 6: Circles 5) If AX = 5 then DX =
Fall 2014 6) Find the perimeter of triangle
ABC.
7) Find the perimeter of the
quadrilateral
We say that a polygon is __________________ ___________ a circle when all vertices (corners) are on
the circle. In this case, we can also say the circle is ______________________ __________ the polygon.
8) Draw a circle inscribed in a square.
9) Draw a circle circumscribed about a pentagon.
Algebra Review: Graph each function and find the key information
f ( x ) = ( x + 1)( x + 5)
Find the x-intercepts (____, ____)
y-intercept (____, ____)
aos
vertex (____, ____)
f ( x ) = ( x − 3)( x + 1)
Find the x-intercepts (____, ____)
y-intercept (____, ____)
aos
vertex (____, ____)
12.2 Chords and Arcs
Chord
Secant
secant line, secant ray, secant segment
5
−3
x+2
4
Find slope
y-intercept (____, ____)
x-intercept (____, ____)
f ( x) =
Unit 6: Circles Fall 2014 II. Discover…..a) Given that ∠AOX ≅ ∠BOX .
a) What can you conclude about ∆AOX and ∆BOX ?Why?
b) What can you conclude about AX and BX ? Why?
Given that AB ≅ XY . Draw triangles with vertex at center. Are the two triangles congruent? Why?
Thm 12.4
Within a circle or congruent circles:
1)
2)
3)
Thm 12.5
Chords that are __________________ from the center are __________
______________ chords are ________________ from the center.
Thm 12.6/7
In a circle, a diameter that is __________________ to a _________ ,
___________ the ____________ and its’ ________.
Converse: ___________________________________
Thm 12.8
Find each of the following:
6
Unit 6: Circles 1)
p = ______
mBC
2)
Fall 2014 p = ______
mBC
3) ON = OM = 8, CM = 7, find
EF =_________
Algebra Review: Solve each system graphically. Identify the solution on the graph.
⎧x + y = 4
⎧x − y = 1
⎧x = 3
1. ⎨
2. ⎨
3. ⎨
⎩ y = −2 x + 3
⎩2 x − 3 y = 5
⎩ y = ( x − 2)( x − 5)
12.3 Inscribed Angles
Inscribed Angle ___________________________
______
__________
Intercepted arc________________________
Thm 12.9
The measure of an inscribed angle = ___________________________
1) Solve for x and y.
2) Solve for x and y.
3) Solve for a and b.
a
60°
30°
b
7
Unit 6: Circles Fall 2014 Corrollaries to Thm 12.9
a) two inscribed angles that intercept ___________________ are __________
b) an angle inscribed in a __________________ is a _____________
c) the ______________________ of a quadrilateral inscribed in a circle are ____________
Thm 12.10 The measure of an angled formed by a _______ and a __________
equals
Find a and b.
Find x and y.
Find w.
200°
w°
12.4 Angle Measures and Segment Lengths
What if the vertex of an angle is no longer located on the circle itself or at the center?
Label as instructed by your teacher.
Theorem 12.11
4.)
5.)
100˚
90⁰
40⁰
6.)
200⁰
x⁰
30º
x
8
x⁰ 25⁰
Unit 6: Circles Fall 2014 Algebra Review: Simplify each rational expression.
1.
x2 − 2x − 3
x2 + 5x + 4
2.
x 2 + 8 x + 16
2x2 + 9x + 4
3.
x2 + 5x
x 2 + 3 x − 10
4.
x2 − 9
3x 2 + 8 x − 3
Theorem 12.12 b 2 secants inside circle
Theorem 12.12 a
2 secant lines
1 secant & 1 tangent
Solve for the missing variable:
1.)
2.)
3.)
4.)
5.)
6.)
7.)
8.)
2
4
9.)
6
y
w
Algebra Review: Solve each quadratic.
1. x 2 − 3 x + 4 = 0
3. x 2 + 5 x + 6 = 0
2. 6 x 2 − 7 x − 5 = 0
4. x 2 + 2 x − 8 = 0
4
3
4
4
6
Sketch a graph for #5.
5. f ( x) = x 2 + 2 x − 8
9
m
3
Unit 6: Circles Fall 2014 12.5 Circles in the Coordinate Plane
Standard Form
For each circle, find the center and the radius. Graph #3.
1. x 2 + y 2 = 9
2. x 2 + y 2 = 16
3. ( x − 3) + ( y − 2 ) = 4
2
2
4. ( x + 1) + ( y − 4 ) = 25
2
2
Graph each circle.
2
2
5. ( x − 3) + ( y + 2 ) = 1
c: ___
_
r: ____
6. x + ( y + 2 ) = 25
2
2
c: ___
_
r: ____
10
Find the equation
7.
c: ___
_
r: ____