Download 9-1 Basic Terms associated with Circles and Spheres

Geo 9
Circles
9-1 Basic Terms associated with Circles and Spheres
1
Circle __________________________________________________________________
Given Point = __________________
Given distance = _____________________
Radius__________________________________________________________________
Chord____________________________________________________________________
Secant___________________________________________________________________
Diameter__________________________________________________________________
Tangent___________________________________________________________________
Point of Tangency___________________________________________________________
Sphere____________________________________________________________________
Label Accordingly:
Congruent circles or spheres__________________________________________________
Concentric Circles___________________________________________________________
Concentric Spheres__________________________________________________________
Inscribed in a circle/circumscribed about the polygon________________________________
_______________________________________
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Geo 9
Circles
SKETCHPAD
2
Geo 9
Circles
9-2 Tangents
3
POWERPOINT
Theorem 9-1 If a line is tangent to a circle , then the line is __________________________
_________________________________.
Corollary: Tangents to a circle from a point are __________________________
A
P
B
Theorem 9-2 If a line in the plane of a circle is perpendicular to a radius at its outer endpoint, then
the line is ________________________.
Inscribed in the polygon/circumscribed about the circle:
look for 2 tangents from the
same point!
A
what if A is a right ange?
Geo 9
Circles
4
Common Tangent ___________________________________________________
Common Internal Tangent
Common External Tangent
Tangent circles ________________________________________________________
Draw the tangent line for each drawing
Name a line that satisfies the given description.
C
F
P
O
A
B
1. Tangent to  P but not to  O. _______
2. Common external tangent to  O and  P. _______
3. Common internal tangent to  O and  P. _______
Geo 9
Circles
4.
5
Circles A, B, C are tangent .
Find the radii of the circles.
AB = 7,
AC = 5
A
x
B
C
5. Find the radius of the circle inscribed in a 3-4-5 triangle.
5
3
4
PP CONCLUSION
CB = 9
Geo 9
Circles
6
6) Circles O and P have radii 18 and 8 respectively. AB is tangent to both circles.
Find AB…………….Hint: connect centers. Find a rt.
A
B
O

P

Geo 9
Circles
9-3 Arcs and Central Angles
7
Central Angle ________________________________________________________
Arc ________________________________________________________________
Measure of a minor arc = ______________
Measure of a major arc = __________ - ______________
Adjacent arcs ____________________
Measure of a semicircle = ___________________
Postulate 16 Arc Addition Postulate: The measure of the arc formed by two adjacent arcs is
_________________________________________.
That is, arcs are additive. Just like with angles, to differentiate an arc from its measure, an “m” must
be included in front of the arc.
Congruent arcs _______________________________
Theorem 9-3 In the same circle or _________________, two minor arcs are _____________ if
_________________________________.
Y
X
R
30
A
O
C
O
Z
W
50
S
1. Name
a) two minor arcs
b) two major arcs
c) a semicircle
d) an acute central angle
e) two congruent arcs
2.
a)
b)
c)
T
Give the measure of each angle or arc:
AC
mWOT
XYT
Geo 9
Circles
8
3. Find the measure of 1 (the central angle)
a)
1

40
b)

1
225
72
130

c)
1
d)
1

30
4. Find the measure of each arc:
A
2x-14
B
4x
2x
C
3x+10
E
3x
D
a) AB
b) BC
5)
c) CD
a) If CB  60 ,
B
A
1
2
O
d) DE
C
e) EA
AO = 10, find <1, <2 and AB
b) If <2 = x find <1, CB
Geo 9
Circles
9
9-4 Arcs and Chords
The arc of the chord is _______________________________________
Theorem 9-5 A diameter that is perpendicular to a chord _______________ the chord and
_________________________.
C
That is, in  O with CD  AB, AZ = BZ and AD  BD
How?
O
Z
A
B
D
Other Theorems: If < AOB = < COD, then what must be
true as well?
A
1)
B
2)
3)
O
C
4)
Geo 9
Circles
Find the following:
10
D
2. x = ______ y = ______ mAB = ______
1. x = ______ y = ______
x
y
5
60
6
13
A
3. MN = ______ KO = ______
y
B
x
4. ACB = ______ mAOC = ______
S
M
N
15
A
K
17
220
C
O
O
B
5. x = ______ y = ______
6. mCD = ______
80
y
O
8
D
x
C
7. CD = 40 , FIND CA
8. If OC = 6, find x and y
40
C
A
C
D
6
O
A
B
60
y
x
A
E
D
B
Geo 9
Circles
11
9-5 Inscribed Angles
By definition, an inscribed angle is an angle whose VERTEX IS ON THE CIRCLE and is
contained in the circle. Inscribed angles can intercept a minor arc or a major arc.
Theorem 9-7 The measure of an inscribed angle is equal to ________________________________
Find angle A and angle B. What generalization can you make?
B
A
C
70 
D
Corollary 1: If two inscribed angles __________________ _____________________________
Corollary 2: An inscribed angle that intercepts a diameter _________________________________
Geo 9
Circles
12
Corollary 3: If a quadrilateral is inscribed in a circle, then its opposite angles are ________________
B
Y
A
X
C
D
Theorem 9-8 the measure of an angle formed by a chord and a tangent is equal to
____________ of the intercepted ___________.
Solve for the variable(s) listed:
60
80
y
x
y
x
80
x
z
z
y
Geo 9
Circles
13
140
y
60
x

y
20
x
50 
20
110
y
x
POWERPOINT
Geo 9
Circles
9-6 Other Angles
14
Sketchpad
Theorem 9-9 The measure of an angle formed by two chords that intersect inside a circle is equal to
1
the sum of the intercepted arcs.
2
x
That is: ____________________
1
y
Theorem 9-10 The measure of an angle formed by secants, two tangents
or a secant and a tangent is equal to ______________________________________
THE VERTEX IS OUTSIDE THE CIRCLE
Case 1
Case 2
Case 3
2 secants
2 tangents
secant/tangent
x
y
y
y
x
x
_________________
_________________
__________________
Geo 9
Circles
15
Given UT is tangent to the circle, mVUT = 30. Find the following:
U
100
W
R
T
V
100
S
1. m WT = ________ 2. mTVS = ________ 3. mRVS = ________ 4. m RS = ________
Given the drawing: AB is tangent to  O; AF is a diameter; m AG = 100, m CE = 30,
m EF = 25. Find the measures of angles 1-8.
C
1=
6
B
2=
E
8
A
4=
O
3
3=
5
F
7
2
1
4
5=
6=
7=
8=
G
Geo 9
Circles
16
ANGLE MEASUREMENT
BASED ON VERTEX
1)
VERTEX AT CENTER
angle = ______________
2)
VERTEX ON CIRCLE
angle = ______________

3)

VERTEX INSIDE CIRCLE
angle = ______________

4)
VERTEX OUTSIDE THE CIRCLE
1
2
SECANT/SECANT
angle = ______________
1
1
2
TANGENT/SECANT
2
TANGENT/TANGENT
Geo 9
Circles
9-7 Circles and Lengths of Segments
17
Theorem 9-11 When two ________ intersect inside a circle, the __________ of the _______
of _______ ____________ equals the ___________ of the ______________
of the ___________ ______________.
That is, in the circle below, given that the two chords intersect, the equation is
t
r
____________ or __________________________
s
u
Theorem 9-12 When two ________ segments are drawn to a circle from an _________
_____________, the product of one secant segment and its __________
______________ is equal to the product of the other secant segment and
its _______________________
That is, in the circle below,
r
_____________ or _______________________________
s
u
t
Theorem 9-13 When a _______ segment and a _________ segment are drawn to a circle
From an ___________ ________ the product of the secant segment and
Its _______ _________ is equal to the __________ of the ____________.
That is, in the circle below:
r
_______________ or ____________________________
s
t
Geo 9
Circles
18
EXAMPLES:
3
x
4
15

4
12
18
x
x
y
9
10
x
4
5
12
3 10
1
2
6
3
3
5
y
x
4
2x
x
y
2
x
4
4
y
7
5
SKETCHPAD
POWERPOINT
Geo 9
Circles
19
40
Find the measure of each numbered angle
given arc measures as indicated.
19
18
17
42 is a central angle
20
16
39
40
14 15
36
20
1
20
2
24
37
23
26
25
41
42
60

38
43
31
27
3
5
28
33
13
34
32
45 44
4
45
35
22 21
30
29
12
11
35
10
6
7
8
9
50
m1__________ m2__________ m3__________
m4___________ m5___________
m6__________ m7__________ m8__________
m9___________ m10__________
m11_________ m12_________ m13_________
m14__________ m15__________
m16_________ m17_________ m18_________
m19__________ m20__________
m21_________ m22_________ m23_________
m24__________ m25__________
m26_________ m27_________ m28_________
m29__________ m30__________
m31_________ m32_________ m33_________
m34__________ m35__________
m36_________ m37_________ m38_________
m39__________ m40__________
m41_________ m42_________ m43_________
m44__________ m45__________
Geo 9
Circles
20
CH 9 CIRCLE REVIEW
(1) Find the measure of each of the numbered
angles, given the figure below with arc
below.
measures as marked. Point O is the center
of the circle.
60
(2) The three circles with centers A , B , and C
are tangent to each other as shown
Find the radius of each circle if AB = 12 ,
AC = 10 and BC = 8.
3
12
C
4
140
O
10

6
8
9
7
40
5
A
B
50
m1 =____ m2 =____ m3 =____ m4 =____
m5 =____ m6 =____ m7 =____ m8 =____
m9 =____ m10 =____
(3) mAB = 120 , AO = 6. Find: AB_____
Circle A_____ , Circle B_____ , Circle C_____
(4) mA = 80 Find: mBDC ______
120
A
B
6
O
B

80
A
O
D
C
(5) BC is tangent to the circle with center O.
AB = 2 , OC = 3. Find: BC______
(6) AB is a diameter, CD  AB , AC = 3 ,
BC = 6. Find: CD______
D
O

2
A
3
B
A
C
3
C
6
B
Geo 9
Circles
21
(7) AE is tangent at B, CD is a diameter,
(8) AB is a diameter, BC is tangent at B,
mA = 40. Find: mBD ____, m EBD____
mAD = 120 , AD = 6 3 .
Find: BC_____, CD_____, OA_____
E

B
O

A
40
A

C
B
D
O
6 3
D
120
C
(9) AB is tangent at A, AF = FD, sides as marked.
marked,
Find: EF______ , AF_______
(10) Given the figure with sides as
Find: BC_______ , EF_______
A
A
4 3
F
3
C
4
E
14
B
C
B
5
10
E
4
6
D
6
F
D
(11) Circles with centers O and P as shown,
OP = 15 , OC = 8 , PD = 4
Find: AB______ , CD_______
(12) Given the figure below with sides as
marked, find the radius of the inscribed
circle________
A
C
20
F
D
A

12

P
O
B
D
C

O
E
B
16
Geo 9
Circles
22
Answers
(1)
m1 = 20 , m2 = 25 , m3 = 55 , m4 = 90
m5 = 25 , m6 = 115 , m7 = 65 , m8 = 115
m9 = 45 , m10 = 130
(2) Circle A = 7 , Circle B = 5 , Circle C = 3
(3) 6 3
(4) mBDC = 260
(5) BC = 4
(6) CD = 3 2
(7) mBD = 130 , mEBD = 65
(8) BC = 4 3 , CD = 2 3 , OA = 6
(9) EF = 9 , AF = 6
(10) BC = 4 , EF = 8
(11) AB = 9 , CD =
(12) 4
209
Geo 9
Circles
23
CH 9 CIRCLES REVIEW II
(1) The circle with center O is inscribed in ABC.
sides as
AC  BC . Find: AC______ , BC_______
(2) CA is tangent to the circle at A,
marked. Find: AC_______
A
A
4
F
6

D
6
E
C

B
C
O
6
O
B
(3) AB is an external tangent segment. Points
O and P are the centers of the circles.
(4) Concentric circles with center O, AC is
tangent to the inner circle, sides as marked.
Find: OB_______ , mADC ________
Find: AB_________
A
8 3
B
B
A
O 
C

P
8
O
4
6



D
D
C
(5) Given the figure below, point O is the center
(6) Given the figure below, mA = 30,
the circle, AC  BD , BD = 26 , AC = 24.
mCFD = 65, BC = DE.
Find: OE_____ , DE_____ , OC______
Find: mCD ____, mBE ____, mBC ____
D
C
A
E
B
C
65
O
A
F
30
E
D
B
Geo 9
Circles
24
(7) The circle below with center O, AC = 12 ,
AC  BD .
Find: OE______ , OC_______DE_______
(8) Given the figure below, DH = HF, with
sides as marked.
Find: GC_______ , DH________
D
C
B
E
A
D
3
C
A
4
G
6
O
H
3
E
120
B
F
(9) The circle with center O is inscribed
in ABC as shown below. AB = AC,
sides as marked. Find: OE_________
(10) Points O and P are the centers of the
circles below. CP = 6
Find: AB_______ , mACB ________
A
A
8
6
O
D
P
E
O
B

F
5
B
C
(11) A chord whose length is 30 is in a circle whose radius is 17. How far is the chord from the
center of the circle?
C
Geo 9
Circles
25
Review Answers II
(1) AC = 6 , BC = 8
(2) AC = 6 3
(3) AB = 4 6
(4) OB = 4 , mADC = 240
(5) OE = 5 , DE = 8 , OC = 13
(6) mCD = 95 , mBE = 35 , mBC = 115
(7) OE = 2 3 , OC = 4 3 , DE = 2 3
(8) GC =
(9) OE =
27
, DH = 3 3
4
10
3
(10) AB = 6 3 , mACB = 240
(11) 8
Geo 9
Circles
26
CH 9 CIRCLES ADDITIONAL REVIEW
1)
Find the radius of a circle in which a 48 cm chord is 8 cm closer to the
center than a 40 cm chord.
AB = 48,
CD = 40
C
D
A
2) In a circle O, PQ = 4
RQ = 10
Q
R
B
PO = 15. Find PS.
P
S
O
3) An isosceles triangle, with legs = 13, is inscribed in a circle. If the altitude
to the base of the triangle is = 5, find the radius of the circle. (There are 2 situations)
13
13
13
Answers:
1) 25
2) 2
3) 16.9
13
Geo 9
Circles
27
SUPPLEMENTARY PROBLEMS CH 9
1) Fill out page one of the Circles Packet.
9.2 TANGENTS
2) A regular polygon is inscribed in a circle so that all vertices of the quadrilateral intersect the circle.
What happens to the regular polygon as the number of sides increases.
3) A circle with a center at (2,1) is tangent to the line y = 3x + 5 at A(-1,2).
Make a sketch in the coordinate plane and draw a radius from the center of the circle to the radius at
point A? Why?
4) In the picture below, AB is a common external tangent. How many common external tangents can
be drawn connecting the 2 circles in each of the following pictures? What shape can be formed if a
radius drawn to a tangent is perpendicular to the tangent?
B
9.3 ARCS AND CENTRAL ANGLES
A
5) If the central angle of a slice of pizza is 36  degrees, how many pieces are in the pizza?
6) Circle O has a diameter DG and central angles COG = 86, DOE = 25, and FOG = 15. Find the
minor arcs CG, CF, EF, and major arc DGF.
7) Draw a circle and label one of its diameters AB. Choose any other point on the circle and call it C.
What can you say about the size of angle ACB? Does it depend on which C you chose? Justify
your response, please.
9.4 ARCS AND CHORDS
Geo 9
Circles
28
D
8) If two chords in the same circle have the same length, then their minor arcs have the same length,
too. True or false? Explain. What about the converse of the statement? Is it true? Why?
9) Draw a circle. Draw two chords of unequal length. Which chord is closer to the center of the circle?
What can be said of the “intercepted arcs”?
10) If P and Q are points on a circle, then the center of the circle must be on the perpendicular
bisector of chord PQ. Explain. Which point on the chord is closest to the center?
Q
9.5 INSCRIBED ANGLES
P
11) The Star Trek Theorem:
a.) Given a circle centered at O, let A,B,and C be points on the circle such
that arc AC is
not equal to arc BC and CL is a diameter. Why must
triangles AOC and AOB be
isosceles?
b) State the pairs of angles that must be congruent in these isosceles triangles.
c) Using EAT, find expressions for the measures of <AOL and <BOL.
d) Based on your statement in part c, explain the statement
<ACL = ½(<AOL) and <OCB = ½(<BOL).
e) Now find an expression for <ACB and simplify to prove that it equals ½<AOB.
C
O
B
A
L