91 Basic Terms associated with Circles and Spheres

Geo 9 1 Circles 9­1 Basic Terms associated with Circles and Spheres

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________________________________ _______________________________________

http://www.pinkmonkey.com/studyguides/subjects/geometry/chap7/g0707101.asp

Geo 9 2 Circles 9­2 Tangents POWERPOINT

Theorem 9­1 If a line is tangent to a circle , then the line is __________________________

_________________________________.

Corollary: Tangents to a circle from a point are __________________________

Theorem 9­2 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:

Common Tangent ___________________________________________________

Common Internal Tangent

Common External Tangent

Tangent circles ________________________________________________________

P

A

B

Geo 9 3 Circles Draw the tangent line for each drawing

Name a line that satisfies the given description.

1. Tangent to ¤ P but not to ¤ O. _______

2. Common external tangent to ¤ O and ¤ P. _______

3. Common internal tangent to ¤ O and ¤ P. _______

In the diagram, ¤M and ¤N are tangent at P. PR and SR are tangents to ¤N. ¤N has a diameter of 16, PQ = 3 and RQ = 12. Complete the following:

4. PM = _____ 5. MQ = _____ 6. PR = _____ 7. SR = _____ 8. NS = _____

9. NR = _____

10. If OR = 6 and TO = 8 then TR = ______, 11. If m∠T = 45 and OT = 4 then TR = _____

A B

O

S

N

R

O T

P Q

M

P

C F

R

Geo 9 4 Circles

11. Circles A, B, C are tangent . AB = 7, AC = 5 CB = 9 Find the radii of the circles.

12. Find the radius of the circle inscribed in a 3­4­5 triangle.

A

B

C

x

3 5

4

Geo 9 5 Circles

13) Circles O and P have radii 18 and 8 respectively. AB suur

is tangent to both circles. Find AB…………….Hint: connect centers. Find a rt.

•

•

A

B

P

O

Geo 9 6 Circles 9­3 Arcs and Central Angles

Central Angle ________________________________________________________ Arc ________________________________________________________________

Identify the central angle, the minor arc, and the major arc of the following circles. Please note how the arc is symbolized!!

If Y and Z are endponts of a diameter, then the two arcs are called ___________________

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 ____________ 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 9­3 In the same circle or _________________, two minor arcs are _____________ if

_________________________________.

1. Name 2. Give the measure of each angle or arc: a) two minor arcs a) AC b) two major arcs b) m∠WOT c) a semicircle c) XYT d) an acute central angle e) two congruent arcs 3. Find the measure of ∠1 (the central angle)

W O

Z

X

O Z

R

S

O

X

Y

Z

T

W

O

30

50

Y

Y

A C

Geo 9 7 Circles a) b)

c) d)

4. Find the measure of each arc:

a) AB b) BC c) CD d) DE e) EA

5) a) If » 60 CB = ° , AO = 10, find <1, <2 and AB

b) If <2 = x find <1, » CB

2x­14

2x

3x+10

3x

4x

A B

C

D

E

A

B

C O

1 2

1

130

1

72

1

40

225

30 1

•

• •

•

Geo 9 8 Circles

9­4 Arcs and Chords

The arc of the chord is _______________________________________

Theorem 9­4 In the same circle or in congruent circles

1. ______________________________________________________

2. ______________________________________________________

Theorem 9­5 A diameter that is perpendicular to a chord _______________ the chord and

_________________________.

That is, in ¤ O with CD ⊥ AB, AZ = BZ and AD ≅ BD

How? HL from congruent triangles.

Theorem 9­6 In the same circle or in congruent circles

1. __________________________________________________________

2. __________________________________________________________

Find the following:

1. x = ______ y = ______ 2. x = ______ y = ______ mAB = ______

D

x y

5

13

A B

O

Z

C

x

y 6 60

B A

Geo 9 9 Circles

3. MN = ______ KO = ______ 4. = ______ m∠AOC = ______

5. x = ______ y = ______ 6. mCD = ______

7. » CD = 40° , FIND » CA 8. If OC = 6, find x and y

C O

A

B

220

M K

15

17

S

N

O

80

O

C

D 8

x

y

A B

C D

40°

A

C

60°

D

B

O

x y

E

6

¼ ACB

Geo 9 10 Circles 9­5 Inscribed Angles POWERPOINT

By definition, an inscribed angle is an angle whose ________ is ______________________. Inscribed angles can intercept a minor arc or a major arc.

Theorem 9­7 The measure of an ___________ angle is equal to _______ the __________

of the ____________ ___________.

Lets draw 3 possibilities:

Case 1 Case 2 Case 3 Center On the inscribed angle Center Inside Center Outside

Corollary 1: If two inscribed angles intercept the ______ arc, then the angles are ______

Corollary 2: An angle ___________ in a ___________ is a _______ angle.

Corollary 3: If a quadrilateral is incribed in a circle, then its _________ angles are _______

Theorem 9­8 the measure of an angle formed by a _________ and a ________ is equal to

____________ the __________ of the intercepted ___________.

Solve for the variable(s) listed:

80

z

x

y 60

x

y

z

80

x

y

Geo 9 11 Circles

60

x

140

y

x

110

20

y

20

x

y

•

50 °

Geo 9 12 Circles 9­6 Other Angles POWERPOINT

Theorem 9­9 The measure of an angle formed by two ________ that intersect inside a

_________ is equal to _______ the ____________ of the ____________

________________________________________________.

That is: ____________________

Theorem 9­10 The measure of an _____ formed by two _____________, two ___________

or a _____________ and a ______________ is equal to ________ the

_______________ of the measures of the ______________ ___________

Case 1 Case 2 Case 3

2 secants 2 tangents secant/tangent

_________________ _________________ __________________

Another way to remember:

x

y

1

y x

x

y x

y

Geo 9 13 Circles

Given UT is tangent to the circle, m∠VUT = 30. Find the following:

1. m ¼ 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 1­8.

1=

2=

3=

4=

5=

6=

7=

8=

F

G

1 2

3

4

5

6

7

C

E

A

B

8 O

R

100

S

V

T

U

100

W

Geo 9 14 Circles

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 angle = ______________

SECANT/SECANT TANGENT/SECANT TANGENT/TANGENT

• •

•

2 1 1 2

1 2

Geo 9 15 Circles 9­7 Circles and Lengths of Segments

Theorem 9­11 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

____________ or __________________________

Theorem 9­12 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,

_____________ or _______________________________

Theorem 9­13 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:

_______________ or ____________________________

r

s

t

t

s

r

u

t

s

r

u

Geo 9 16 Circles EXAMPLES:

x

10

12

3

15

4

x

x

x

y

4 5

9

y

1

3 3

x

2x

2

y

4

2 x

4

5

7

4

12 18

x

•

x

y

3 10

5

4

6

Geo 9 17 Circles

Find the measure of each numbered angle given arc measures as indicated.

∠42 is a central angle

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__________

8 9

10

11 12

13 43 44 45

35°

15

16 17 18 19

20

22

23

24

25

26

27

29 30

31

32

33 34

36

37

38

39 40

41

1 2

3

4 5

6

14

45° 21

28

35

42

7

40°

60° •

50°

20°

20°

Geo 9 18 Circles

CH 9 CIRCLE REVIEW

(1) Find the measure of each of the numbered (2) The three circles with centers A , B , and C angles, given the figure below with arc are tangent to each other as shown

below. measures as marked. Point O is the center Find the radius of each circle if AB = 12 , of the circle. AC = 10 and BC = 8.

m∠1 =____ m∠2 =____ m∠3 =____ m∠4 =____ m∠5 =____ m∠6 =____ m∠7 =____ m∠8 =____ Circle A_____ , Circle B_____ , Circle C_____ m∠9 =____ m∠10 =____

(3) » mAB= 120° , AO = 6. Find: AB_____ (4) m∠A = 80° Find: ¼ mBDC______

(5) BC is tangent to the circle with center O. (6) AB is a diameter, CD AB ⊥ , AC = 3 , AB = 2 , OC = 3. Find: BC______ BC = 6. Find: CD______

• O

60°

5 6

7

8

9

10 140°

50°

1 2 3

4 40°

O

A

B

120°

6

B

A

C

O A

B

•

C

• D 80°

3

O

A

B

•

C

2

3 6 A B C

D

Geo 9 19 Circles

(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 = 3 6 .

Find: BC_____, CD_____, OA_____

(9) AB is tangent at A, AF = FD, sides as marked. (10) Given the figure with sides as marked,

Find: EF______ , AF_______ Find: BC_______ , EF_______

(11) Circles with centers O and P as shown, (12) Given the figure below with sides as OP = 15 , OC = 8 , PD = 4 marked, find the radius of the inscribed Find: AB______ , CD_______ circle________

3 6

O A B •

C D 120°

O A

B

40° • C D

E •

O

B

•

A

C

D

P •

16

12

20

O •

A

B C

D

E

F

A

C

D

E F B

4 3

3 4 14

D

B A

C

E 6

6

10 5

4

F

Geo 9 20 Circles

Answers

(1) m∠1 = 20 o , m∠2 = 25 o , m∠3 = 55 o , m∠4 = 90 o

m∠5 = 25 o , m∠6 = 115 o , m∠7 = 65 o , m∠8 = 115 o

m∠9 = 45 o , m∠10 = 130 o

(2) Circle A = 7 , Circle B = 5 , Circle C = 3

(3) 6 3

(4) ¼ mBDC = 260 o

(5) BC = 4

(6) CD = 3 2

(7) » mBD = 130 o , m EBD = 65 ∠ o

(8) BC = 4 3 , CD = 2 3 , OA = 6

(9) EF = 9 , AF = 6

(10) BC = 4 , EF = 8

(11) AB = 9 , CD = 209

(12) 4

Geo 9 21 Circles

CH 9 CIRCLES REVIEW II

(1) The circle with center O is inscribed in ∆ABC. (2) CA is tangent to the circle at A, sides as

AC BC ⊥ . Find: AC______ , BC_______ marked. Find: AC_______

(3) AB is an external tangent segment. Points (4) Concentric circles with center O, AC is O and P are the centers of the circles. tangent to the inner circle, sides as marked.

Find: AB_________ Find: OB_______ , ¼ mADC ________

(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____

6

4

O •

A

B C

D

E

F

• D

P •

A

O •

• C

B

4 6

30° 65° A

B C

D

E

F O

A

B

C

D

E

•

6

O

A

B •

C 6

8

3 8

O

A B C

• D

•

Geo 9 22 Circles

(7) The circle below with center O, AC = 12 , (8) Given the figure below, DH = HF, with

AC BD ⊥ . sides as marked. Find: OE______ , OC_______DE_______ Find: GC_______ , DH________

(9) The circle with center O is inscribed (10) Points O and P are the centers of the in ∆ABC as shown below. AB = AC, circles below. CP = 6

sides as marked. Find: OE_________ Find: AB_______ , ¼ mACB________

(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?

B

120°

O

A C

D

E

• A

B C D

E

F

G H

3

4 6

3

B

8

5

C

O

A

D E

F

•

6

• • O P

A

B

• C

Geo 9 23 Circles

Review Answers II

(1) AC = 6 , BC = 8

(2) AC = 6 3

(3) AB = 4 6

(4) OB = 4 , ¼ mADC = 240 o

(5) OE = 5 , DE = 8 , OC = 13

(6) » mCD = 95 o , » mBE = 35 o , » mBC = 115 o

(7) OE = 2 3 , OC = 4 3 , DE = 2 3

(8) GC = 27 4

, DH = 3 3

(9) OE = 10 3

(10) AB = 6 3 , ¼ mACB = 240 o

(11) 8

Geo 9 24 Circles

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

2) In a circle O, PQ = 4 RQ = 10 PO = 15. Find PS.

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)

Answers:

1) 25 2) 2 3) 16.9

A B

C D

R

S O

P Q

13 13

13 13

Geo 9 25 Circles

SUPPLEMENTARY PROBLEMS CH 9 1) 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.

2) An arc is a piece of the circle that has length and degree measure as well. What is the angular size of an arc that a diameter intercepts? What is this arc called?

3) What is the radius of the smallest circle that surrounds a 5 by 12 rectangle?

4) Two circles of radius 10 cm are drawn so that their centers are 12 cm apart. The points of intersection of the circles determine a segment defined as a common chord. What kind of quadrilateral is formed when the centers of the circles are connected to the endpoints of the chord? What special property do its diagonals have?

5) 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 point of tangency A( ­1,2) . What is the angle of intersection between the tangent and the radius at point A? Why?

6) In the picture below, AB is a common external tangent and CD is a common internal tangent. How many common external tangents can be drawn connecting the 2 circles in each of the following pictures? Pg. 335/classroom ex.

A

B

C

D

Geo 9 26 Circles 7) In the picture below with the common external tangent PQ and circles with centers at B and A, what kind of quadrilateral is PABQ?

8) If the central angle of a slice of pizza is 36° degrees, how many pieces are in the pizza?

9) 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.

10) 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.

11) 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?

12) 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”?

13) 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?

14) A circular park 80 meters in diameter has a straight path (AB) cutting across it. It is 24 meters from the center of the park to the closest point on this path. How long is the path?

B

Q

P

A

Q

P

B A

Geo 9 27 Circles

15) A 20 inch chord is drawn in a circle with a 20 inch radius. What is the angular size of the minor arc of the chord?

16) 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.

17) Segment AB, which is 25 inches long, is the diameter of a circle. Chord PQ meets AB perpendicularly at C, where AC = 16 in. Find the length of PQ.

18) A circle goes through the points A, B, C, and D consecutively. The chords AC and BD intersect at P. Draw AB and DC. What can you say about triangles ABP and DCP? Why?

19) A piece of broken circular gear is brought into a metal shop so that a replacement can be built. A ruler is placed across two points on the rim, and the length of the chord is found to be 14 inches. The distance from the midpoint of this chord to the nearest point on the rim is found to be 4 inches. Find the radius of the original gear.

B

L

C

O

A