Circles



Circles

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|Day |Topic |Assignment |Completed |

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|Day 1 |Circles and Circumference |D1 HW – SP 10-1 | |

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|Day 2 |Measures of Arcs and Angles |D2 HW – Pg. 9 | |

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|Day 3 |Arcs and Chords |D3 HW – SG 10-3 | |

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|Day 4 |Arcs and Chords |D4 HW – SP 10-3 | |

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|Day 5 |Inscribed Angles |D5 HW – Pg. 19 | |

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|Day 6 |Inscribed Angles |D6 HW – SP 10-4 | |

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|Day 7 |Tangents and Secants |D7 HW – SP 10-5 (1, 2, 5, 6) | |

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|Day 8 |Tangents and Secants |D8 HW – Pg. 26 | |

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|Day 9 |Angle Relationships in Circles |D9 HW – Pg. 29 and SP 10-6 (1 – 6) | |

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|Day 10 |Angle Relationships in Circles |D10 HW – Pg. 32 and | |

| | |SP 10-6 (7 – 12) | |

| |Segment Relationships in Circles | | |

|Day 11 | |D11 HW - Pg. 35 | |

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|Day 12 |Segment Relationships in Circles |D12 HW – Pg. 39 and rest of SP 10-7 if not| |

| | |finished in class. | |

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|Day 13 |Equations of Circles |D13 HW – Pg. 43 | |

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|Day 14 |Review |Study | |

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|Day 15 |TEST |Good Luck! | |

Circles and Circumference

A circle is the set of points in a plane equidistant, called the radius, from a given point called the center of the circle.

Naming a Circle:

Name the circle according to the center point.

The name of the circle is circle P or • P.

( P

Segments that intersect a circle have special names.

|Segments that Intersect Circles |

|A radius (plural radii) is a segment with endpoints at the center| |

|and on the circle. | |

|A chord is a segment with endpoints on the circle. Chords do not | |

|go through the center of the circle. | |

|A diameter is a segment that passes through the center of the | |

|circle and has endpoints on the circle. (Collinear radii.) | |

Use the diagram to answer the questions.

1. _____ is the name of the circle.

2. _____ is the diameter.

3. _____ is a chord.

4. _____, _____, and _____ are the radii.

5. All the radii are _____ to each other.

|Radius and Diameter Relationships |

|Radius Formula: |Diameter Formula: |

|r = [pic] or r = [pic]d |d = 2r |

|***All radii in a circle are (. | |

1. If the radius of a circle is 10, what is the diameter?

2. If the diameter of a circle is 14, what is the radius?

Use the diagram to answer the following questions.

3. If TV = 8, what is the UQ?

4. If QU = 12, what is QT?

5. If QU = 30, what is:

a. QT?

b. TV?

c. TU?

|Circle Pairs |

|Congruent Circles – two circles are congruent if they have |Concentric Circles – coplanar circles that have the same center. |

|congruent radii. | |

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|( ( |( |

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|GH ( JK, so • G ( • J |• A with radius AB and • A with radius AC are concentric. |

Two circles can intersect in two different ways:

|2 Points of Intersection |1 Point of Intersection |No Points of Intersection |

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1. If RT = 21, what is the length of QV?

•

2. The radius of • S is 15 units, the radius of • R is 10 units, and DS = 9 units. Find:

a. CD = _____

b. RC = _____

3. The diameter of • Y is 22 units, the diameter of • X is 16 units, and WZ = 5 units. Find XY.

Circumference

1. Find the circumference of a circle whose diameter is 10 cm. to the nearest tenth of a cm.

2. Find the exact circumference of a circle whose diameter is 4 in.

3. Find the exact circumference of a circle whose radius is 7 units.

4. Find the diameter of a circle to the nearest hundredth if the circumference is 106.4 in.

5. Find the diameter of a circle to the nearest tenth if the circumference is 65.4 ft.

A polygon is inscribed in a circle if all the vertices lie on the circle.

A circle is circumscribed about a polygon if it contains all the vertices of the polygon.

6. A square with side length of 9 in. is inscribed in • J. Find the exact circumference.

7. A square with side length of 10 ft. is inscribed in • J. Find the exact circumference.

Measuring Angles and Arcs

|Arcs and Arc Measures |

| Minor Arc | |

|A minor arc is the shortest arc connecting two endpoints on a | |

|circle. | |

|It’s measure is less than 180. | |

|It is equal to the central angle. | |

|Major Arc | |

|A major arc is the longest arc connecting two endpoints on a | |

|circle. | |

|major arc = 360 – x. | |

|Semicircle (half circle) | |

|A semicircle is an arc with endpoints that lie on a diameter. |m[pic] = 180 |

|It is equal to 180. |m[pic] = 180 |

| |Each half circle = 180. |

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Identify each arc as a major arc, minor arc or semicircle of the circle.

1. m[pic] 2. m[pic] 3. m[pic]

|Central Angles |

|Central Angle | |

|A central angle is an angle with its vertex in the center of the | |

|circle. | |

|It is equal to the arc it intercepts. | |

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| |If m[pic]= 93, then m(DFC = 93. |

|Sum of Central Angles | |

|The sum of the all measures of the central angles in a circle is | |

|equal to 360. |m(1 + m(2 + m(3 = 360 |

| |• |

4. m[pic] 5. m[pic] 6. m[pic]

7. PR and QT are diameters. Find the measure of each arc.

a. m[pic] b. m[pic] c. m[pic]

d. m[pic] e. m[pic] f. m[pic]

g. m[pic] h. m[pic]

Find the value of x.

1. 2. 3.

4. 5. 6.

7. Find the value of x.

8. In the accompanying diagram of circle O, [pic] has a measure of 200°. What is the m(BOA?

9. If m[pic]BOC = 15, find:

a. m(AOC

b. m[pic]

c. m[pic]

d. m[pic]

e. m[pic]

10. In circle O, AB is a diameter and m(AOC = 100. Find:

a. m(COB

b. m[pic]

c. m[pic]

d m[pic]

e. m[pic]

f. m[pic]

11. Find the value of x.

D2 HW - Arcs and Central Angles

1. In circle O, m[pic]AOB = 87, m[pic]BOC = 93, and m[pic]COD = 35. Find the measure of each of the following:

a. [pic]DOA b. [pic] c. [pic]

d. [pic] e. [pic] f. [pic]

g. [pic] h. [pic] i. [pic]

2. Lines AB and CD intersect at O, the center of the circle, and m[pic]AOC = 25. Find the measure of each of the following:

a. [pic]COB b. [pic]BOD c. [pic]DOA

d. [pic] e. [pic] f. [pic]

g. [pic] h. [pic] i. [pic]

3. In circle O, [pic]AOC and [pic]COB are supplementary. If m[pic]AOC = 2x,

m[pic]COB = x + 90, and m[pic]AOD = 3x + 10, find:

a. x b. m[pic]AOC c. m[pic]COB

d. m[pic]AOD e. m[pic]DOB f. m [pic]

g. m[pic] h. m[pic] i. m[pic]

j. m[pic] k. m[pic] l. m[pic]

Arcs and Chords

Congruent arcs have the same measure.

|Congruent Arcs, Chords, and Central Angles |

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|If m(BEA ( m(CED, then BA ( CD. |If BA ( CD, then |If [pic] ( [pic], then m(BEA ( m(CED. |

| |[pic] ( [pic]. | |

|Congruent central angles have congruent |Congruent chords have congruent arcs. |Congruent arcs have congruent central |

|chords. | |angles. |

1. The m(BEA = 82 and the measure of chord CD = 10.

a. What is the m(CED?

b. What is the measure of chord BA?

c. What is the m(BEC?

2. The measure of chord CD = 8 and m[pic] = 76.

a. What is the measure of chord BA?

b. What is the m[pic]?

3. The m[pic] = 86.

a. What is the m[pic]?

b. What is the m(CED?

c. What is the m(BEA?

4. QR ( ST. Find m[pic]?

5. The m(HLG ( m(KLJ. Find m[pic].

6. Find the value of x.

a. b.

.

c. d. • J ( • K

|Bisecting Arcs and Chords |

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|In a circle, if a radius or a diameter is | |

|perpendicular (() to a chord, then it | |

|bisects the chord and its arcs. | |

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|Note: | |

|You can use the Pythagorean Theorem to find| |

|the lengths. By drawing in a radius, you | |

|create a right triangle. | |

|BE, ED, and EA are radii. |Since EA ( BD, then AE bisects BD |

| |Therefore, BC ( CD and [pic] ( [pic]. |

1. In • S, m[pic]= 98. Find m[pic].

2. In • S, TR = 6. Find PR.

3. In • S, if diameter QST ( chord PR at V. If PR = 18, find PV and VR.

4. In • J, GH ( KM. If JL = 8 and KM = 30, find the length of KJ.

5. In • J, GH = 34 and KM = 30. Find the length of JL.

6. Find TV to the nearest tenth.

|Chords Equidistant from Center |

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|In the same circle or in congruent circles,| |

|two chords are congruent if and only if | |

|they are equidistant from the center. | |

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| |If FG ( JH, then LX = LY. |

1. In • A, WX = XY = 22, BE = 5x, and CE = 3x + 4. Find EB.

2. In • H, PQ = 3x – 4 and RS = 14. Find x.

|Inscribed Angle Theorem |

|Inscribed Angle | |

|An inscribed angle is an angle with its vertex on the circle. | |

|It is equal to ½ the arc it intercepts. | |

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|Note: | |

|The arc is equal to 2 times the angle. | |

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| |If m[pic]= 100, then m(DFC = 50. |

|Inscribed Angles |

|If inscribed angles of a circle intercept the same arc, then the |An inscribed angle intercepts a diameter or semicircle if and only of the angle |

|angles are congruent. |is a right angle. |

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| |If [pic] is a semicircle (180), then m(C = 90. |

|(ABC and (ADC intercept [pic], so that (ABC = (ADC. |If m(C = 90, then [pic] = 180 (semicircle) and AB is a diameter. |

1. Find m(N. 2. Find m(N. 3. Find m(N.

4. Find m[pic] 5. Find m[pic]. 6. Find m[pic].

7. Find m(LMP and m[pic]. 8. Find m(GFJ and m[pic].

9. Find m(P and m[pic]. 10. Find m(C and m[pic].

11. Find m(ACB and m(AOB. 12. Find m(C and m[pic].

13. Find:

a. m[pic] b. m[pic] c. m[pic]

d. m(ABE e. m(AOC f. m(BAC

14. Find x. 15. Find m(FJH.

16. Find m(T . 17. Find m(F.

18. In the diagram of circle O, diameters AOB and COD are drawn as well as chords AC, CB, and DB. If m(ACD = 32 and [pic] = 116, find:

a. m[pic] b. m[pic] c. m[pic]

d. m(AOC e. m(AB f. m(CBD

g. m(BOD h. m(ACB

|Inscribed Angle Theorem |

|If a quadrilateral is inscribed in a circle, then its opposite | |

|angles are supplementary. | |

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|(A and (C are supplementary. | |

|m(A + m(C = _____. | |

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|(B and (D are supplementary. | |

|m(B + m(D = _____. | |

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| |ABCD is inscribed in • E. |

1. Find the value of x. 2. Find m(B.

3. Find all the angle measures of each quadrilateral.

a. b.

x = _____ x = _____

m(R = _____ m(J = _____

m(S = _____ m(K = _____

m(T = _____ m(L = _____

m(V = _____ m(M = _____

D5 HW - Inscribed Angle

1. Find each measure.

a. m(N. b. m(N. c. m(N.

d. m[pic] e. m[pic]. f. m[pic].

g. Find x. h. m(H and m(G i. m(T and m(V

Tangents and Secants

|Lines That Intersect Circles |

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|A secant is a line or ray that intersects a circle at | |

|two points. | |

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|A tangent is a line or ray that intersects the circle at| |

|exactly one point, called the point of tangency. | |

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|Tangent Circles |

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|Two coplanar circles that intersect at exactly one point| |

|are called tangent circles. | |

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1. Identify each line or segment that intersects each circle.

a. b.

Line l is a ____________. Line NM is a ____________.

Line m is a ____________. Ray LM is a ____________.

FG is a ____________. KJ is a ____________.

2. Use circle P to identify each line, segment or point.

a. secant line _____

b. point of tangency _____

c. tangent line _____

d. chord _____

e. a point in the exterior of the circle _____

f. a point in the interior of the circle _____

3. In each figure, draw the common tangents. If no common point exists, state no common tangent.

a. b. c.

The shortest distance from a tangency to the center of a circle is the radius drawn to the point of tangency.

|Angle Formed by a Tangent and a Radius Theorem |

|A line is a tangent to a circle if it is perpendicular (() to a | |

|radius drawn to the point of tangency. | |

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1. EA is a __________.

2. Line BAC is a _______________ line.

3. A is the point of _______________.

4. (EAC and (EAB are __________ angles.

5. m(EAC = _____ and m(EAB = _____.

6. Tangent BA intersects radius OA. What is m(OAB?

7. JK is a radius of • J. Determine whether KL is a tangent to • J. Justify your answer.

a. b.

8. JH is tangent to • G. Find the value of x.

a. b.

9. BC is tangent to • A. Find the value of x to the nearest tenth.

More than one line can be tangent to a circle.

|Congruent Tangents |

|If two segments are tangent to a circle from the same external | |

|point, then the segments are congruent. | |

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|EF and EG are tangent to • C. | |

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| |FE ( GE |

1. AC and BC are tangent to • D. Find the value of x.

2. QR and SR are tangent to • T. Find the value of x.

3. WX and YX are tangent to • Z. Find the value of WX and YX.

|Circumscribed Polygons |Circumscribed NOT Polygons |

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|A polygon is circumscribed about a circle if every side of the | |

|polygon is tangent to the circle. | |

1. Triangle ABC circumscribes circle G. Find:

a. AE

b. DC

c. CF

d. perimeter of (ABC

2. Quadrilateral RSTU is circumscribed about circle J. If the perimeter is 18 units, find x.

.

3. Triangle JKL is circumscribe about circle R. Find:

a. x.

b. the perimeter of (JKL.

c. What type of triangle is (JKL?

Vocabulary

point of tangency exterior tangent line secant

chord interior perpendicular tangent

D8 HW

1. Find the value of x to the nearest tenth.

a. b.

2. Find the value of x.

a. b.

3. Triangle ABC is circumscribed about circle G. Find:

a. x.

b. the perimeter of (ABC.

Angle Relationships in Circles

|Intersections On or Inside a Circle |

|If a tangent or a secant intersects on a circle | |

|at the point of tangency then the angle formed is|m(ABC = [pic] m[pic] |

|half the measure of its intercepted arc. |m(ABC = [pic](150) |

|***Tangent BC and secant BA intersect at B. |m(ABC = 75 |

|If two secants or chords intersect in the | m(1 = [pic] (m[pic] + m[pic]) |

|interior of a circle, then the measure of the |m(1 = [pic](84 + 130) |

|angle formed is half the sum of the measures of |m(1 = [pic](214) |

|its intercepted arcs. |m(1 = 107 |

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|***Chords AB and CD intersect at E. | |

1. Find each measure.

a. m(QPR b. m(QPS and m(RPS

c. m(RQS if m[pic]= 238 d. m[pic]

e. m[pic] f. m[pic]

2. Find x.

a. b.

c. d.

e. f.

g. h.

D9 HW

Find each measure.

a. m(3 b. m(JMK c. m[pic]

d. m[pic] e. m(QPS f. m(ADB

If two segments intersect in the exterior of a circle, then the measure of the angle formed is half the difference of the measures of its intercepted arcs.

|A Tangent and a Secant |Two Tangents |Two Secants |

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|m(1 = [pic] (m[pic] - m[pic]) |m(2 = [pic] (m[pic] - m[pic]) |m(3 = [pic] (m[pic] - m[pic]) |

|m(1 = [pic](210 – 102) |m(2 = [pic](262 – 98) |m(3 = [pic](164 – 76) |

|=[pic](108) = 54 |= [pic](164) = 82 |= [pic](88) = 44 |

|m(1 = 54 |m(2 = 82 |m(3 = 44 |

1. Find the value of x.

a. b.

c. d.

e. f.

g. h.

i. j.

k. l. Find m(VTU

D10 HW

Find each measure.

1. m(F 2. m(S 3. m(R

4. m[pic] 5. m[pic]

Segment Relationships in Circles

|Chord-Chord Product Theorem |

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|If two chords intersect in the interior of a | |

|circle then the products of the lengths of the | |

|segments of the chords are equal. | |

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| |AE ( EB = CE ( ED |

| |x ( 6 = 9 ( 4 |

| |6x = 36 |

| |x = 6 |

| |AE = 6 |

1. Find x.

a. b.

c. d.

e. f

2. In circle O, chords AC and BD intersect at E. If EC = 8, BE = 9, and ED = 12, find AE.

D11 HW

Find x. Round answers to the nearest tenth if necessary.

a. b. c.

d. e. f.

Segments Intersecting Outside the Circle

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|A secant segment is a segment of a secant with at least one | |

|endpoint on the circle. | |

|An external secant segment is the part of the secant segment | |

|that lies in the exterior of the circle. | |

|A tangent segment is a segment of a tangent with one endpoint | |

|on the circle. | |

If two segments intersect outside a circle, then the following theorems are true.

|Secant- Secant Product Theorem | |

|The products of the lengths of one secant segment and its | |

|external segment equals the product of the lengths of the other | |

|secant segment and its external segment. | |

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|whole ( outside = whole ( outside | |

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| |AE ( BE = CE ( DE |

|Secant- Tangent Product Theorem | |

|The products of the lengths the secant segment and its external | |

|segment equals the lengths of the tangent segment squared. | |

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|whole ( outside = tangent2 | |

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| |AE ( BE = DE2 |

1. Find x.

a. b.

c. d.

e. f.

g. h.

Segment Relationships in Circles

D12 HW

Find each measure.

1. 2.

3. 4.

5. 6.

Equations of Circles

1. Write the equation of each circle.

a. a circle with center (5, 10) and radius 6

b. a circle with center (8, -4) and radius 3

c. a circle with center (-3, 12) and radius 5

d. a circle with center (-7, -8) and radius [pic]

e. a circle with center (0, 15) and radius 2.5

f. a circle with center (18, 0) and radius 10

g. a circle with center (0, 0) and radius 9

Identify the center and the radius from the equation.

Given the equation (x – 1)2 + (y + 4)2 = 9, state the center and the radius.

(x – h)2 + (y – k)2 = r2

(x – 1)2 + (y + 4)2 = 9

(x – 1)2 + (y – (- 4))2 = 32

Therefore, center = (1, -4) and the radius = 3.

*****Shortcut to get the center of the circle take the numbers out and change the signs.

2. For each equation, state the center and the radius of the circle.

a. (x - 6)2 + ( y – 4)2 = 25

b. (x - 3)2 + ( y – 7)2 = 16

c. (x + 4)2 + ( y – 5)2 = 36

d. (x + 1)2 + ( y – 5)2 = 14

e. x2 + ( y – 2)2 = 20

f. (x + 12) + y2 = 1

g. x2 + y2 = 12

3. Graph each circle.

a. a circle with center (-1, 3) and radius 9

b. (x + 2)2 + ( y – 3)2 = 25

center = _______

radius = _____

c. (x - 4)2 + y2 = 9

center = _______

radius = _____

D13 HW

1. Write the equation of each circle.

a. a circle with center (1, 3) and radius 8

b. a circle with center (2, -5) and radius 1

c. a circle with center (0, 15) and radius 11

d. a circle with center (-9, -4) and radius [pic]

2. For each equation, state the center and the radius of the circle.

a. (x - 8)2 + ( y – 1)2 = 144

b. (x - 3)2 + y2 = 36

c. (x + 4)2 + ( y – 6)2 = 81

d. (x + 7)2 + ( y + 7)2 =100

4. Graph each circle.

a. a circle with center (3, -4) and radius 4

You can also write the equation of a circle when you know the center and one point on the circle.

To write the equation of the circle,

➢ Use the distance formula to find the radius.

➢ Write the equation of the circle.

1. Write the equation of circle L that has center L(3, 7) and passes through (1,7).

2. Write the equation of circle B that has center (5, 4) and passes through (-3,4).

3. Graph and write the equation of the circle with a center (-2, 4) and passes through point (-6, 7).

If the diameter of a circle is perpendicular to a chord, it __________ the chord.

3. In circle O, diameter AOB ( chord CD at E. If CD = 16, find the length of CE.

4. Find x.

a. b.

5. In circle F, diameter DFE ( chord AC at B. Find the length of BF.

6. In circle O, diameter MON ( chord QR at T. If MN = 34 and QR = 30, find the length of OT.

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V

F

E

D

C

B

A

AB is a chord.

EC is a diameter.

EF, FC, and FD are all radii. All the radii are (.

U

T

H

Q

G

N

M

L

P

J

K

B

C

A

3

R

S

V

Q

R

T

S

C

D

1

6

•J

Y

W

Z

X

Circumference – Distance around a circle.,

C = (d

Chocolate pie’s(() delicious.

R

•J

A

2

Q

X

B

Y

5

F

E

D

C

93

E

CED is a major arc.

m[pic] = 360 – 93 = 267

CD is a minor arc. m[pic] = 93

130

A

D

B

A

C

•

50

100

30

C

E

50

R

D

B

40

A

S

40

S

T

P

Q

R

U

93

(DFC is a central angle.

m(DFC = 93

D

C

F

6

14

x

145

Chords XY and QR intersect at S.

x

3

x

100

A

B

D

F

175

x

Z

40

4

85

140

x

112

x

x

33

C

R

O

C

A

B

V

B

A

C

O

U

T

8x - 4

13x - 3

5x + 5

20x

Q

12

S

Y

F

A

X

x

B

A

D

C

E

E

C

D

A

B

E

C

D

A

B

E

C

D

A

B

G

x

x

F

E

E

C

D

A

B

E

C

D

A

B

4x

S

T

Q

R

3x + 22

x + 13

2x + 5

J

K

G

H

L

2x + 3

8x

S

T

Q

R

L

120

x

F

G

E

D

2x + 1

F

N

5

U

M

•O

•J

•K

3x - 7

Q

P

L

80

x

J

K

G

H

P

T

S

R

Q

E

C

D

A

B

2x + 15

V

3x - 5

S

T

X

6

P

T

S

R

Q

P

T

S

R

Q

V

K

H

J

M

G

L

K

H

J

M

G

L

S

U

6

V

R

T

H

G

J

F

Y

E

Y

C

B

W

3x + 4

5x

142

X

R

H

U

8

T

P

S

Q

8

F

C

D

(DFC is an inscribed angle.

m(DFC = ½ (100)

= 50

100

B

x

A

E

8

D

C

B

D

A

72

A

C

B

100

A

C

B

113

Q

S

P

80

L

M

N

R

62

L

K

N

35

L

K

N

N

48

L

M

P

36

36

G

110

F

H

J

56

O

70

M

P

N

40

E

98

C

D

F

50

B

80

C

88

A

B

124

C

O

A

E

R

5x + 8

Q

S

P

H

5x

F

J

G

4x + 9

4x + 2

H

G

J

9x - 3

116

32

O

D

C

B

A

C

A

E

D

B

E

3x + 5

H

F

G

4x

x

A

2x - 30

D

B

C

7x

R

8x + 8

V

S

T

11x

2x + 2

x - 17

J

x + 25

M

K

L

A

C

B

l

C is the point of tangency.

l is a tangent.

AB is a secant.

points of tangency

X

2x + 9

H

Y

F

G

m

W

Y

J

L

M

K

N

l

Z

k

P

z

X

B

R

A

t

l

E

X

C

Y

D

A tangent line is _______________ to the radius of a circle drawn to the point of tangency. A line that is perpendicular to the radius of a circle at a point on the circle is a ______________________ __________ to the circle.

The _____________ of a circle is the set of points outside the circle.

Ex: Pt. _____

The _________________ is the point where the tangent and the circle intersect.

Ex: Pt. _____

The _____________ of a circle is the set of points inside the circle.

Ex: Pt. _____

A _____________ is a line that intersects the circle at exactly one point.

Ex: Line _____

A _____________ is a segment whose endpoints lie on the circle.

Ex: Segment _____

A _____________ is a line that intersects a circle at two points.

Ex: Line _____

C

A

E

B

B

E

A

C

B

O

A

K

J

L

9

8

15

8

6

12

K

J

L

12

x

8

J

G

2

x

4

J

G

H

H

B

17

x

14

C

A

2x - 5

A

D

B

E

F

G

C

external point

x + 15

C

R

3x + 8

S

T

Q

26

3x + 6

B

86

C

O

A

G

A

B

C

7

8

E

B

F

D

10

U

C

R

D

T

x

A

S

x

3

3

4x - 9

M

O

N

7

x + 3

J

L

K

R

C

B

1

A

D

A

U

S

R

179

T

M

J

3

K

L

N

G

H

2

F

E

71

102

210

76

164

98

232

G

C

x

F

E

C

150

A

C

B

E

1

D

B

A

130

84

148

Q

S

R

P

T

P

R

S

Q

250

R

S

Q

E

F

C

D

64

L

K

J

H

116

A

B

C

D

108

A

C

B

E

143

D

x

75

110

x

H

K

G

J

97

L

47

x

J

L

K

G

H

116

55

x

N

P

75

M

Q

128

x

Y

W

V

Z

88

154

x

F

E

H

G

76

96

x

Y

U

W

V

62

60

x

J

M

L

K

25

B

9

4

E

D

6

x

A

E

B

D

AE is a secant segment.

BE is an external secant segment.

x

74

S

R

140

T

U

ED is a tangent segment..

x

29

M

P

75

Q

L

N

F

x

26

K

G

B

H

J

88

x

56

D

C

95

N

A

W

x

68

Y

Z

224

110

F

25

x

K

G

J

H

62

x

M

P

141

Q

L

N

35

x

M

P

108

Q

L

Q

30

57

P

V

81

U

S

R

T

Angle Relationships in Circles

Vertex lies _____ the circle.

Vertex lies __________ a circle.

Vertex lies __________ a circle.

Angle measure is half the measure of the intercepted arc.

Angle measure is half the difference of the measures of the intercepted arcs.

Angle measure is half the sum of the measures of the intercepted arcs.

X

Z

Y

85

30

B

C

A

84

D

C

A

B

E

60

m(AEB = m(AEB =[pic](___ + ___) = ____

m(XZY = [pic](___)

= ____

76

m(ACB = [pic](___ - ___)

= ____

xaaaaaaaaaaaa

12aaaaaaaaaaaa

10aaaaaaaaaaaa

5aaaaaaaaaaaa

Caaaaaaaaaaaa

Daaaaaaaaaaaa

Baaaaaaaaaaaa

Eaaaaaaaaaaaa

Aaaaaaaaaaaaa

Aaaaaaaaaaaaa

Eaaaaaaaaaaaa

Baaaaaaaaaaaa

Daaaaaaaaaaaa

Caaaaaaaaaaaa

6aaaaaaaaaaaa

15aaaaaaaaaaaa

x

2aaaaaaaaaaaa

E

M

Jaaaaaaaaaaaa

Naaaaaaaaaaaa

Laaaaaaaaaaaa

3aaaaaaaaaaaa

4

12

x

Eaaaaaaaaaaaa

A

Caaaaaaaaaaaa

Daaaaaaaaaaaa

Baaaaaaaaaaaa

6

3

8

xaaaaaaaaaaaa

Paaaaaaaaaaaa

M

Kaaaaaaaaaaaa

Laaaaaaaaaaaa

Naaaaaaaaaaaa

x + 8

x

x + 10

x + 1

Paaaaaaaaaaaa

M

Kaaaaaaaaaaaa

Laaaaaaaaaaaa

Naaaaaaaaaaaa

x + 2

x

x + 12

x + 6

O

T

Q

M

R

N

F

E

B

2.5

C

A

D

4.8

2.5

x

7

2

2

x

10

x

O

E

A

C

A

D

B

C

B

E

D

D

E

B

G

A

L

x

8

K

10

6

J

H

T

S

5

4

V

9

x

R

U

I

S

10

24

G

8

x

E

H

F

B

D

A

7.5

4.5

C

5

x

E

R

P

R

12

P

S

8

x

Q

x

S

6

4

Q

F

H

12

J

4

x

G

M

K

12

L

x + 2

4

J

Secants AB and DB intersect at B.

Secant AY and tangent YX intersect at X.

p ( p = p ( p

RS ( SQ = XS ( SY

O ( W = tan2

XF ( XA= XY2

O ( W = O ( W

BZ ( BA = BF ( BD

___ ( ___ = ___ ( ___

___ = ___

___ = ___

SR = ___

___ ( ______ = ___ ( ___

__________ = ___

___ = ___

___ = ___

ZA = ___

___ ( ___ = ___

___ = ___

___ = ___

XY = ___

y

x

y

y

x

y

y

x

y

y

x

y

y

x

y

Example

What is the equation of a circle whose center is (2, -3) and has a radius of 2?

(x – h)2 + (y – k)2 = r2

(x – 2)2 + (y – (-3))2 = 22

(x – 2)2 + (y + 3)2 = 4

Graph

Equation

The equation of a circle with center (h, k) and radius r is

(x – h)2 + (y – k)2 = r2

(h, k) is the __________ of the circle, r is the __________ of the circle, and (x, y) is a __________ on the circle.

Circles in the Coordinate Plane

X

4x + 3

Y

Z

W

5x + 1

94

x

F

E

H

G

68

2x

27

D

F

E

x

44

S

R

124

T

U

14

17

C

B

A

G

R

2x

S

T

Q

6x - 4

X

5x - 9

Y

Z

W

x + 7

H

12

x

20

J

G

B

11

x

7

C

B

x

41

D

C

A

M

K

6

L

103

A

W

x

56

x

4

J

M

K

Y

Z

190

106

G

x

F

E

x

43

S

R

103

14

L

x

7

J

M

K

9

L

7

y

x

y

x

5

J

F

I

S

x

3

G

2

12

E

H

B

D

A

3

x

C

4

8

E

B

D

A

5

S

T

6x - 2

5x + 4

H

32

F

J

G

48

A

C

B

R

118

Q

S

P

T

U

x

88

S

R

152

T

U

E

F

C

D

72

P

R

S

Q

110

x

C

7

10

E

5

x

6

12

3

x

6

10

x

x + 20

x + 10

A

B

C

D

154

R

P

Q

N

74

51

90

3

2x

R

x

Q

S

P

4x

45

R

2x + 30

V

102

A

C

B

92

L

M

N

51

L

K

N

124

210

L

K

N

x + 5

74

L

M

J

K

H

79

77

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