INSCRIBED ANGLES Find the indicated measure. m G FDG120870

VOCABULARY Copy and complete: The points A and B are on (C. If C is a point on }AB, then }AB is a ? .

WRITING Explain how you can determine from the context whether the words radius and diameter are referring to a segment or a length.

MATCHING TERMS Match the notation with the term that best describes it. C E D A H G A. Center B. Radius C. Chord D. Diameter E. Secant F. Tangent G. Point of tangency H. Common tangent B

MATCHING TERMS Match the notation with the term that best describes it. C E D A H G A. Center B. Radius C. Chord D. Diameter E. Secant F. Tangent G. Point of tangency H. Common tangent ] BH

MATCHING TERMS Match the notation with the term that best describes it. C E D A H G A. Center B. Radius C. Chord D. Diameter E. Secant F. Tangent G. Point of tangency H. Common tangent AB

MATCHING TERMS Match the notation with the term that best describes it. C E D A H G A. Center B. Radius C. Chord D. Diameter E. Secant F. Tangent G. Point of tangency H. Common tangent .] AB

MATCHING TERMS Match the notation with the term that best describes it. C E D A H G A. Center B. Radius C. Chord D. Diameter E. Secant F. Tangent G. Point of tangency H. Common tangent ] AE

MATCHING TERMS Match the notation with the term that best describes it. C E D A H G A. Center B. Radius C. Chord D. Diameter E. Secant F. Tangent G. Point of tangency H. Common tangent G

MATCHING TERMS Match the notation with the term that best describes it. C E D A H G A. Center B. Radius C. Chord D. Diameter E. Secant F. Tangent G. Point of tangency H. Common tangent CD

MATCHING TERMS Match the notation with the term that best describes it. C E D A H G A. Center B. Radius C. Chord D. Diameter E. Secant F. Tangent G. Point of tangency H. Common tangent BD

ERROR ANALYSIS Describe and correct the error in the statement about the diagram. A D B 6 9 E The length of secant }AB is 6.

COORDINATE GEOMETRY Use the diagram at the right. C D 3 6 9 369 What are the radius and diameter of (C?

COORDINATE GEOMETRY Use the diagram at the right. C D 3 6 9 369 What are the radius and diameter of (D?

COORDINATE GEOMETRY Use the diagram at the right. C D 3 6 9 369 Copy the circles. Then draw all the common tangents of the two circles.

DRAWING TANGENTS Copy the diagram. Tell how many common tangents the circles have and draw them.

DRAWING TANGENTS Copy the diagram. Tell how many common tangents the circles have and draw them.

DRAWING TANGENTS Copy the diagram. Tell how many common tangents the circles have and draw them.

DETERMINING TANGENCY Determine whether }AB is tangent to (C. Explain. C 5 A B

DETERMINING TANGENCY Determine whether }AB is tangent to (C. Explain. B 9 15 A 18

DETERMINING TANGENCY Determine whether }AB is tangent to (C. Explain. 0 A B 48 5

ALGEBRA Find the value(s) of the variable. In Exercises 2426, B and D are points of tangency. C 24 r r 16

ALGEBRA Find the value(s) of the variable. In Exercises 2426, B and D are points of tangency. C 9 r r 6

ALGEBRA Find the value(s) of the variable. In Exercises 2426, B and D are points of tangency. C r r 7 1

ALGEBRA Find the value(s) of the variable. In Exercises 2426, B and D are points of tangency. B D A 3x 1 10 7x 2 6

ALGEBRA Find the value(s) of the variable. In Exercises 2426, B and D are points of tangency. 13 D B A 2

ALGEBRA Find the value(s) of the variable. In Exercises 2426, B and D are points of tangency. A 4x 2 1 B 3x2 1 4x 2 4

COMMON TANGENTS A common internal tangent intersects the segment that joins the centers of two circles. A common external tangent does not intersect the segment that joins the centers of the two circles. Determine whether the common tangents shown are internal or external.

COMMON TANGENTS A common internal tangent intersects the segment that joins the centers of two circles. A common external tangent does not intersect the segment that joins the centers of the two circles. Determine whether the common tangents shown are internal or external.

MULTIPLE CHOICE In the diagram, (P and (Q are tangent circles. }RS is a common tangent. Find RS. A 22} 15 B 4 C 2} 15 D 8 P R S 3 5

REASONING In the diagram, ] PB is tangent to (Q and (R. Explain why }PA > }PB > }PC even though the radius of (Q is not equal to the radius of (R. P R P B A

TANGENT LINES When will two lines tangent to the same circle not intersect? Use Theorem 10.1 to explain your answer.

ANGLE BISECTOR In the diagram at right, A and D are points of tangency on (C. Explain how you know that ] BC bisects ABD. (Hint: Use Theorem 5.6, page 310.) C D A B

SHORT RESPONSE For any point outside of a circle, is there ever only one tangent to the circle that passes through the point? Are there ever more than two such tangents? Explain your reasoning.

CHALLENGE In the diagram at the right, AB 5 AC 5 12, BC 5 8, and all three segments are tangent to (P. What is the radius of (P? B C E A D F

BICYCLES On modern bicycles, rear wheels usually have tangential spokes. Occasionally, front wheels have radial spokes. Use the definitions of tangent and radius to determine if the wheel shown has tangential spokes or radial spokes.

BICYCLES On modern bicycles, rear wheels usually have tangential spokes. Occasionally, front wheels have radial spokes. Use the definitions of tangent and radius to determine if the wheel shown has tangential spokes or radial spokes.

GLOBAL POSITIONING SYSTEM (GPS) GPS satellites orbit about 11,000 miles above Earth. The mean radius of Earth is about 3959 miles. Because GPS signals cannot travel through Earth, a satellite can transmit signals only as far as points A and C from point B, as shown. Find BA and BC to the nearest mile. MI !

SHORT RESPONSE In the diagram, }RS is a common internal tangent (see Exercises 2728) to (A and (B. Use similar triangles to explain why } AC BC 5 } RC SC A CB S R

PROVING THEOREM 10.1 Use parts (a)(c) to prove indirectly that if a line is tangent to a circle, then it is perpendicular to a radius. GIVEN c Line m is tangent to (Q at P. PROVE c m }QP a. Assume m is not perpendicular to }QP. Then the perpendicular segment from Q to m intersects m at some other point R. Because m is a tangent, R cannot be inside (Q. Compare the length QR to QP. b. Because }QR is the perpendicular segment from Q to m, }QR is the shortest segment from Q to m. Now compare QR to QP. c. Use your results from parts (a) and (b) to complete the indirect proof. m P R P

PROVING THEOREM 10.1 Write an indirect proof that if a line is perpendicular to a radius at its endpoint, the line is a tangent. GIVEN c m }QP PROVE c Line m is tangent to (Q. m P P

PROVING THEOREM 10.2 Write a proof that tangent segments from a common external point are congruent. GIVEN c }SR and }ST are tangent to (P. PROVE c }SR > }ST Plan for Proof Use the HypotenuseLeg Congruence Theorem to show that nSRP > nSTP. P R T S

CHALLENGE Point C is located at the origin. Line l is tangent to (C at (24, 3). Use the diagram at the right to complete the problem. a. Find the slope of line l. b. Write the equation for l. c. Find the radius of (C. d. Find the distance from l to (C along the y-axis. (24, 3) C l

D is in the interior of ABC. If m ABD 5 258 and m ABC 5 708, find m DBC. (p. 24)

Find the values of x and y. (p. 154) x 8 508 y 8

Find the values of x and y. (p. 154) 1028 x 8 3y 8

Find the values of x and y. (p. 154) (2x 1 3)8 (4y 2 7)8 1378

A triangle has sides of lengths 8 and 13. Use an inequality to describe the possible length of the third side. What if two sides have lengths 4 and 11? (p. 328)

VOCABULARY Copy and complete: If ACB and DCE are congruent central angles of (C, thenCAB andCDE are ? .

WRITING What do you need to know about two circles to show that they are congruent? Explain.

MEASURING ARCS }AC and}BE are diameters of (F. Determine whether the arc is a minor arc, a major arc, or a semicircle of (F. Then find the measure of the arc. A F B D C E 708 458 CBC

MEASURING ARCS }AC and}BE are diameters of (F. Determine whether the arc is a minor arc, a major arc, or a semicircle of (F. Then find the measure of the arc. A F B D C E 708 458 CDC

MEASURING ARCS }AC and}BE are diameters of (F. Determine whether the arc is a minor arc, a major arc, or a semicircle of (F. Then find the measure of the arc. A F B D C E 708 458 CDB

MEASURING ARCS }AC and}BE are diameters of (F. Determine whether the arc is a minor arc, a major arc, or a semicircle of (F. Then find the measure of the arc. A F B D C E 708 458 AE

MEASURING ARCS }AC and}BE are diameters of (F. Determine whether the arc is a minor arc, a major arc, or a semicircle of (F. Then find the measure of the arc. A F B D C E 708 458 AD

MEASURING ARCS }AC and}BE are diameters of (F. Determine whether the arc is a minor arc, a major arc, or a semicircle of (F. Then find the measure of the arc. A F B D C E 708 458 ABC

MEASURING ARCS }AC and}BE are diameters of (F. Determine whether the arc is a minor arc, a major arc, or a semicircle of (F. Then find the measure of the arc. A F B D C E 708 458 ACD

MEASURING ARCS }AC and}BE are diameters of (F. Determine whether the arc is a minor arc, a major arc, or a semicircle of (F. Then find the measure of the arc. A F B D C E 708 458 CEAC

MULTIPLE CHOICE In the diagram, }QS is a diameter of (P. Which arc represents a semicircle? A CQR B CRQT C CQRS D CQRT P T P

CONGRUENT ARCS Tell whether the red arcs are congruent. Explain why or why not. B C D 1808 708 408

CONGRUENT ARCS Tell whether the red arcs are congruent. Explain why or why not. L M N 858

CONGRUENT ARCS Tell whether the red arcs are congruent. Explain why or why not. 928 X Y Z 928

ERROR ANALYSIS Explain what is wrong with the statement. You cannot tell if (C > (D because the radii are not given.

ARCS Two diameters of (P are }AB and }CD . If mCAD 5 208, find mCACD and mCAC .

MULTIPLE CHOICE (P has a radius of 3 andCAB has a measure of 908. What is the length of }AB ? A 3} 2 B 3} 3 C 6 D 9 A B P

SHORT RESPONSE On (C, mCEF 5 1008, mCFG 5 1208, and mCEFG 5 2208. If H is on (C so that mCGH 5 1508, explain why H must be onCEF

REASONING In (R, mCAB 5 608, mCBC 5 258, mCCD 5 708, and mCDE 5 208. Find two possible values for mCAE

CHALLENGE In the diagram shown, }PQ }AB , }QA is tangent to (P, and mCAVB 5 608. What is mCAUB ? B V U P P

CHALLENGE In the coordinate plane shown, C is at the origin. Find the following arc measures on (C. a. mCBD b. mCAD c. mCAB y B(4, 3) A(3, 4) C D(5, 0)

BRIDGES The deck of a bascule bridge creates an arc when it is moved from the closed position to the open position. Find the measure of the arc.

DARTS On a regulation dartboard, the outermost circle is divided into twenty congruent sections. What is the measure of each arc in this circle?

EXTENDED RESPONSE A surveillance camera is mounted on a corner of a building. It rotates clockwise and counterclockwise continuously between Wall A and Wall B at a rate of 108 per minute. a. What is the measure of the arc surveyed by the camera? b. How long does it take the camera to survey the entire area once? c. If the camera is at an angle of 858 from Wall B while rotating counterclockwise, how long will it take for the camera to return to that same position? d. The camera is rotating counterclockwise and is 508 from Wall A. Find the location of the camera after 15 minutes.

CHALLENGE A clock with hour and minute hands is set to 1:00 P.M. a. After 20 minutes, what will be the measure of the minor arc formed by the hour and minute hands? b. At what time before 2:00 P.M., to the nearest minute, will the hour and minute hands form a diameter?

Determine if the lines with the given equations are parallel. (p. 180) y 5 5x 1 2, y 5 5(1 2 x)

Determine if the lines with the given equations are parallel. (p. 180) 2y 1 2x 5 5, y 5 4 2 x

Trace nXYZ and point P. Draw a counterclockwise rotation of nXYZ 1458 about P. (p. 598) X Z Y

Find the product. (p. 641) (x 1 2)(x 1 3)

Find the product. (p. 641) (2y 2 5)(y 1 7)

Find the product. (p. 641) (x 1 6)(x 2 6)

Find the product. (p. 641) (z 2 3)2

Find the product. (p. 641) (3x 1 7)(5x 1 4)

Find the product. (p. 641) (z 2 1)(z 2 4)

VOCABULARY Describe what it means to bisect an arc.

WRITING Two chords of a circle are perpendicular and congruent. Does one of them have to be a diameter? Explain your reasoning.

FINDING ARC MEASURES Find the measure of the red arc or chord in (C. A B E C D 758

FINDING ARC MEASURES Find the measure of the red arc or chord in (C. A D 1288 C B 34 34

FINDING ARC MEASURES Find the measure of the red arc or chord in (C. F G C E J H

ALGEBRA Find the value of x in (Q. Explain your reasoning. B 4x P 3x 1 7

ALGEBRA Find the value of x in (Q. Explain your reasoning. 5x 2 6 2x 1 9

ALGEBRA Find the value of x in (Q. Explain your reasoning. R T P 908 8x 2 13 6x 1 9

ALGEBRA Find the value of x in (Q. Explain your reasoning. A D B 5x 2 7 1

ALGEBRA Find the value of x in (Q. Explain your reasoning. A D B 6 22 6 P 3x

ALGEBRA Find the value of x in (Q. Explain your reasoning. E H A B F P 4x 1 1 x 1 8

REASONING In Exercises 1214, what can you conclude about the diagram shown? State a theorem that justifies your answer. B E C A

REASONING In Exercises 1214, what can you conclude about the diagram shown? State a theorem that justifies your answer. F H J P

REASONING In Exercises 1214, what can you conclude about the diagram shown? State a theorem that justifies your answer. N L R S M P

MULTIPLE CHOICE In the diagram of (R, which congruence relation is not necessarily true? A }PQ >}QN B }NL > }LP C CMN >CMP D }PN > }PL P L R M

ERROR ANALYSIS Explain what is wrong with the diagram of (P. C F D G P H B A 6 6 7 7

ERROR ANALYSIS Explain why the congruence statement is wrong. BC >CCD C D E B A

IDENTIFYING DIAMETERS Determine whether}AB is a diameter of the circle. Explain your reasoning. C D 4 6 6 9

IDENTIFYING DIAMETERS Determine whether}AB is a diameter of the circle. Explain your reasoning. C D

IDENTIFYING DIAMETERS Determine whether}AB is a diameter of the circle. Explain your reasoning. C D E 3 3 5

REASONING In the diagram of semicircleCQCR , PC} }AB and mCAC 5 308. Explain how you can conclude that nADC nBDC. P P R D C A

WRITING Theorem 10.4 is nearly the converse of Theorem 10.5. a. Write the converse of Theorem 10.5. Explain how it is different from Theorem 10.4. b. Copy the diagram of (C and draw auxiliary segments }PC and }RC. Use congruent triangles to prove the converse of Theorem 10.5. c. Use the converse of Theorem 10.5 to show that QP 5 QR in the diagram of (C. P S Q C R T

ALGEBRA In (P below, }AC , }BC , and all arcs have integer measures. Show that x must be even. P C B A

CHALLENGE In (P below, the lengths of the parallel chords are 20, 16, and 12. Find mCAB . P B A

LOGO DESIGN The owner of a new company would like the company logo to be a picture of an arrow inscribed in a circle, as shown. For symmetry, she wantsCAB to be congruent toCBC. How should }AB and }BC be related in order for the logo to be exactly as desired?

OPEN-ENDED MATH In the cross section of the submarine shown, the control panels are parallel and the same length. Explain two ways you can find the center of the cross section.

PROVING THEOREM 10.3 In Exercises 27 and 28, prove Theorem 10.3. D C A B P GIVEN c }AB and }CD are congruent chords. PROVE c CAB >CCD

PROVING THEOREM 10.3 In Exercises 27 and 28, prove Theorem 10.3. D C A B P GIVEN c }AB and }CD are chords andCAB >CCD. PROVE c }AB > }CD

CHORD LENGTHS Make and prove a conjecture about chord lengths. a. Sketch a circle with two noncongruent chords. Is the longer chord or the shorter chord closer to the center of the circle? Repeat this experiment several times. b. Form a conjecture related to your experiment in part (a). c. Use the Pythagorean Theorem to prove your conjecture.

MULTI-STEP PROBLEM If a car goes around a turn too quickly, it can leave tracks that form an arc of a circle. By finding the radius of the circle, accident investigators can estimate the speed of the car. a. To find the radius, choose points A and B on the tire marks. Then find the midpoint C of }AB. Measure }CD, as shown. Find the radius r of the circle. b. The formula S 5 3.86} fr can be used to estimate a cars speed in miles per hours, where f is the coefficient of friction and r is the radius of the circle in feet. The coefficient of friction measures how slippery a road is. If f 5 0.7, estimate the cars speed in part (a).

PROVING THEOREMS 10.4 AND 10.5 Write proofs. GIVEN c }QS is the perpendicular bisector of }RT. PROVE c }QS is a diameter of (L. Plan for Proof Use indirect reasoning. Assume center L is not on }QS . Prove that nRLP > nTLP, so }PL }RT . Then use the Perpendicular Postulate. R P P S

PROVING THEOREMS 10.4 AND 10.5 Write proofs. GIVEN c }EG is a diameter of (L. EG} }DF PROVE c }CD > }CF ,CDG >CFG Plan for Proof Draw }LD and }LF . Use congruent triangles to show CD}> }CF and DLG > FLG. Then showCDG >CFG . G F C

PROVING THEOREM 10.6 For Theorem 10.6, prove both cases of the biconditional. Use the diagram shown for the theorem on page 666.

CHALLENGE A car is designed so that the rear wheel is only partially visible below the body of the car, as shown. The bottom panel is parallel to the ground. Prove that the point where the tire touches the ground bisectsCAB .

The measures of the interior angles of a quadrilateral are 1008, 1408, (x 1 20)8, and (2x 1 10)8. Find the value of x. (p. 507)

Quadrilateral JKLM is a parallelogram. Graph ~JKLM. Decide whether it is best described as a rectangle, a rhombus, or a square. (p. 552) J(23, 5), K(2, 5), L(2, 21), M(23, 21)

Quadrilateral JKLM is a parallelogram. Graph ~JKLM. Decide whether it is best described as a rectangle, a rhombus, or a square. (p. 552) J(25, 2), K(1, 1), L(2, 25), M(24, 24)

VOCABULARY Copy and complete: If a circle is circumscribed about a polygon, then the polygon is ? in the circle.

WRITING Explain why the diagonals of a rectangle inscribed in a circle are diameters of the circle.

INSCRIBED ANGLES Find the indicated measure. m A A B C 848

INSCRIBED ANGLES Find the indicated measure. m G F D G 1208 70

INSCRIBED ANGLES Find the indicated measure. m N M 1608

INSCRIBED ANGLES Find the indicated measure. mCRS R S 67

INSCRIBED ANGLES Find the indicated measure. mCVU T V U 308

INSCRIBED ANGLES Find the indicated measure. mCWX W X 1108 758

ERROR ANALYSIS Describe the error in the diagram of (C. Find two ways to correct the error. Q R S C 45 100

CONGRUENT ANGLES Name two pairs of congruent angles. C D A B

CONGRUENT ANGLES Name two pairs of congruent angles. L K M J

CONGRUENT ANGLES Name two pairs of congruent angles. Z Y W

ALGEBRA Find the values of the variables. R S T P y 8 808 x8 958

ALGEBRA Find the values of the variables. D G F m8 2k8 608 60

ALGEBRA Find the values of the variables. 548 4b8 1308 1108

MULTIPLE CHOICE In the diagram, ADC is a central angle and m ADC 5 608. What is m ABC? A 158 B 308 C 608 D 1208 C A B

INSCRIBED ANGLES In each star below, all of the inscribed angles are congruent. Find the measure of an inscribed angle for each star. Then find the sum of all the inscribed angles for each star.

MULTIPLE CHOICE What is the value of x? A 5 B 10 C 13 D 15 F G (8x 1 10)8 (12x 1 40)8

PARALLELOGRAM Parallelogram QRST is inscribed in (C. Find m R.

REASONING Determine whether the quadrilateral can always be inscribed in a circle. Explain your reasoning. Square

REASONING Determine whether the quadrilateral can always be inscribed in a circle. Explain your reasoning. Rectangle

REASONING Determine whether the quadrilateral can always be inscribed in a circle. Explain your reasoning. Parallelogram

REASONING Determine whether the quadrilateral can always be inscribed in a circle. Explain your reasoning. Kite

REASONING Determine whether the quadrilateral can always be inscribed in a circle. Explain your reasoning. Rhombus

REASONING Determine whether the quadrilateral can always be inscribed in a circle. Explain your reasoning. Isosceles trapezoid

CHALLENGE In the diagram, C is a right angle. If you draw the smallest possible circle through C and tangent to }AB, the circle will intersect }AC at J and }BC at K. Find the exact length of }JK 5 3

ASTRONOMY Suppose three moons A, B, and C orbit 100,000 kilometers above the surface of a planet. Suppose m ABC 5 908, and the planet is 20,000 kilometers in diameter. Draw a diagram of the situation. How far is moon A from moon C?

CARPENTER A carpenters square is an L-shaped tool used to draw right angles. You need to cut a circular piece of wood into two semicircles. How can you use a carpenters square to draw a diameter on the circular piece of wood?

WRITING A right triangle is inscribed in a circle and the radius of the circle is given. Explain how to find the length of the hypotenuse.

PROVING THEOREM 10.10 Copy and complete the proof that opposite angles of an inscribed quadrilateral are supplementary. GIVEN c (C with inscribed quadrilateral DEFG PROVE c m D 1 m F 5 1808, m E 1 m G 5 1808. By the Arc Addition Postulate, mCEFG 1 ? 5 3608 and mCFGD 1 mCDEF 5 3608. Using the ? Theorem, mCEDG 5 2m F, mCEFG 5 2m D, mCDEF 5 2m G, and mCFGD 5 2m E. By the Substitution Property, 2m D 1 ? 5 3608, so ? . Similarly, ? . F G C

PROVING THEOREM 10.7 If an angle is inscribed in (Q, the center Q can be on a side of the angle, in the interior of the angle, or in the exterior of the angle. In Exercises 3133, you will prove Theorem 10.7 for each of these cases. Case 1 Prove Case 1 of Theorem 10.7. GIVEN c B is inscribed in (Q. Let m B 5 x8. Point Q lies on }BC . PROVE c m B 5 1 }2 mCAC Plan for Proof Show that nAQB is isosceles. Use the Base Angles Theorem and the Exterior Angles Theorem to show that m AQC 5 2x8. Then, show that mCAC 5 2x8. Solve for x, and show that m B 5 1 }2 mCAC C B P A

PROVING THEOREM 10.7 If an angle is inscribed in (Q, the center Q can be on a side of the angle, in the interior of the angle, or in the exterior of the angle. In Exercises 3133, you will prove Theorem 10.7 for each of these cases. Case 2 Use the diagram and auxiliary line to write GIVEN and PROVE statements for Case 2 of Theorem 10.7. Then write a plan for proof. B C P A

PROVING THEOREM 10.7 If an angle is inscribed in (Q, the center Q can be on a side of the angle, in the interior of the angle, or in the exterior of the angle. In Exercises 3133, you will prove Theorem 10.7 for each of these cases. Case 3 Use the diagram and auxiliary line to write GIVEN and PROVE statements for Case 3 of Theorem 10.7. Then write a plan for proof. D B P A C

PROVING THEOREM 10.8 Write a paragraph proof of Theorem 10.8. First draw a diagram and write GIVEN and PROVE statements.

PROVING THEOREM 10.9 Theorem 10.9 is written as a conditional statement and its converse. Write a plan for proof of each statement.

EXTENDED RESPONSE In the diagram, (C and (M intersect at B, and }AC is a diameter of (M. Explain why ] AB is tangent to (C. CM A B

CHALLENGE In Exercises 37 and 38, use the following information. You are making a circular cutting board. To begin, you glue eight 1 inch by 2 inch boards together, as shown at the right. Then you draw and cut a circle with an 8 inch diameter from the boards. F H L FH is a diameter of the circular cutting board. Write a proportion relating GJ and JH. State a theorem to justify your answer.

CHALLENGE In Exercises 37 and 38, use the following information. You are making a circular cutting board. To begin, you glue eight 1 inch by 2 inch boards together, as shown at the right. Then you draw and cut a circle with an 8 inch diameter from the boards. F H L Find FJ, JH, and JG. What is the length of the cutting board seam labeled }GK ?

SPACE SHUTTLE To maximize thrust on a NASA space shuttle, engineers drill an 11-point star out of the solid fuel that fills each booster. They begin by drilling a hole with radius 2 feet, and they would like each side of the star to be 1.5 feet. Is this possible if the fuel cannot have angles greater than 458 at its points? 1.5 ft 2 ft

Find the approximate length of the hypotenuse. Round your answer to the nearest tenth. (p. 433) 60 5

Find the approximate length of the hypotenuse. Round your answer to the nearest tenth. (p. 433) 82 38 x

Find the approximate length of the hypotenuse. Round your answer to the nearest tenth. (p. 433) 26 16

Graph the reflection of the polygon in the given line. (p. 589) y-axis x 1 1 A C

Graph the reflection of the polygon in the given line. (p. 589) x 5 3 y x 1 1 E H F G

Graph the reflection of the polygon in the given line. (p. 589) y 5 2 y x 1 1 S

Sketch the image of A(3, 24) after the described glide reflection. (p. 608) Translation: (x, y) (x, y 2 2) Reflection: in the y-axis

Sketch the image of A(3, 24) after the described glide reflection. (p. 608) Translation: (x, y) (x 1 1, y 1 4) Reflection: in y 5 4x

VOCABULARY Copy and complete: The points A, B, C, and D are on a circle and ] AB intersects ] CD at P. If m APC 5 1 }2 1mCBD 2 mC AC 2, then P is ? (inside, on, or outside) the circle.

WRITING What does it mean in Theorem 10.12 if mCAB 5 08? Is this consistent with what you learned in Lesson 10.4? Explain your answer.

FINDING MEASURES Line t is tangent to the circle. Find the indicated measure. mCAB B A t 658

FINDING MEASURES Line t is tangent to the circle. Find the indicated measure. mCDEF E F D t 1178

FINDING MEASURES Line t is tangent to the circle. Find the indicated measure. m 1 t 2608

MULTIPLE CHOICE The diagram at the right is not drawn to scale. }AB is any chord that is not a diameter of the circle. Line m is tangent to the circle at point A. Which statement must be true? A x 90 B x 90 C x 5 90 D x 90 B A m x8

FINDING MEASURES Find the value of x. 1458 858 x8 A C B

FINDING MEASURES Find the value of x. x8 F G E

FINDING MEASURES Find the value of x. J K L M (2x 2 30)

FINDING MEASURES Find the value of x. P 2478

FINDING MEASURES Find the value of x. F G D E 298 1

FINDING MEASURES Find the value of x. 348 (x 1 6)8 (3x 2 2)8

MULTIPLE CHOICE In the diagram, l is tangent to the circle at P. Which relationship is not true? A m 1 5 1108 B m 2 5 708 C m 3 5 808 D m 4 5 908 T R P P l 808 608 1

ERROR ANALYSIS Describe the error in the diagram below. A D C E F 60 50

SHORT RESPONSE In the diagram at the right, ] PL is tangent to the circle and }KJ is a diameter. What is the range of possible angle measures of LPJ? Explain. L J P K

CONCENTRIC CIRCLES The circles below are concentric. a. Find the value of x. 408 1108 x8 b. xpress c in terms of a and b. a8 c8 b

INSCRIBED CIRCLE In the diagram, the circle is inscribed in nPQR. Find mCEF , mCFG , and mCGE P R 408 608 808

ALGEBRA In the diagram, ] BA is tangent to (E. Find mCCD . D C A B 408 3x8 7

WRITING Points A and B are on a circle and t is a tangent line containing A and another point C. a. Draw two different diagrams that illustrate this situation. b. Write an equation for mCAB in terms of m BAC for each diagram. c. When will these equations give the same value for mCAB ?

CHALLENGE Find the indicated measure(s). Find m P if mCWZY 5 2008. X P Y W

CHALLENGE Find the indicated measure(s). Find mCAB and mCED G A F B C 1158 208

VIDEO RECORDING In the diagram at the right, television cameras are positioned at A, B, and C to record what happens on stage. The stage is an arc of (A. Use the diagram for Exercises 2224. Find m A, m B, and m C.

VIDEO RECORDING In the diagram at the right, television cameras are positioned at A, B, and C to record what happens on stage. The stage is an arc of (A. Use the diagram for Exercises 2224. The wall is tangent to the circle. Find x without using the measure of C.

VIDEO RECORDING In the diagram at the right, television cameras are positioned at A, B, and C to record what happens on stage. The stage is an arc of (A. Use the diagram for Exercises 2224. You would like Camera B to have a 308 view of the stage. Should you move the camera closer or further away from the stage? Explain.

HOT AIR BALLOON You are flying in a hot air balloon about 1.2 miles above the ground. Use the method from Example 4 to find the measure of the arc that represents the part of Earth that you can see. The radius of Earth is about 4000 miles.

EXTENDED RESPONSE A cart is resting on its handle. The angle between the handle and the ground is 148 and the handle connects to the center of the wheel. What are the measures of the arcs of the wheel between the ground and the cart? Explain. 14

PROVING THEOREM 10.11 The proof of Theorem 10.11 can be split into three cases. The diagram at the right shows the case where }AB contains the center of the circle. Use Theorem 10.1 to write a paragraph proof for this case. What are the other two cases? (Hint: See Exercises 3133 on page 678.) Draw a diagram and write plans for proof for the other cases. B C A P

PROVING THEOREM 10.12 Write a proof of Theorem 10.12. GIVEN c Chords }AC and }BD intersect. PROVE c m 1 5 1 }2 1mCDC 1 mCAB 2 B C D A

PROVING THEOREM 10.13 Use the diagram at the right to prove Theorem 10.13 for the case of a tangent and a secant. Draw }BC . Explain how to use the Exterior Angle Theorem in the proof of this case. Then copy the diagrams for the other two cases from page 681, draw appropriate auxiliary segments, and write plans for proof for these cases. A C B

PROOF Q and R are points on a circle. P is a point outside the circle. }PQ and }PR are tangents to the circle. Prove that }QR is not a diameter.

CHALLENGE A block and tackle system composed of two pulleys and a rope is shown at the right. The distance between the centers of the pulleys is 113 centimeters and the pulleys each have a radius of 15 centimeters. What percent of the circumference of the bottom pulley is not touching the rope?

Classify the dilation and find its scale factor. (p. 626) C 12 16 P9 P

Classify the dilation and find its scale factor. (p. 626) 9 C 15 P

Use the quadratic formula to solve the equation. Round decimal answers to the nearest hundredth. (pp. 641, 883) x2 1 7x 1 6 5 0

Use the quadratic formula to solve the equation. Round decimal answers to the nearest hundredth. (pp. 641, 883) x2 2 x 2 12 5 0

Use the quadratic formula to solve the equation. Round decimal answers to the nearest hundredth. (pp. 641, 883) x2 1 16 5 8x

Use the quadratic formula to solve the equation. Round decimal answers to the nearest hundredth. (pp. 641, 883) x2 1 6x 5 10

Use the quadratic formula to solve the equation. Round decimal answers to the nearest hundredth. (pp. 641, 883) 5x 1 9 5 2x2

Use the quadratic formula to solve the equation. Round decimal answers to the nearest hundredth. (pp. 641, 883) 4x2 1 3x 2 11 5 0

VOCABULARY Copy and complete: The part of the secant segment that is outside the circle is called a(n) ? .

WRITING Explain the difference between a tangent segment and a secant segment.

FINDING SEGMENT LENGTHS Find the value of x. 12 10

FINDING SEGMENT LENGTHS Find the value of x. 18 x 2 3 10

FINDING SEGMENT LENGTHS Find the value of x. x 8 x 1 8

FINDING SEGMENT LENGTHS Find the value of x. x 6 8 1

FINDING SEGMENT LENGTHS Find the value of x. 7 4 5

FINDING SEGMENT LENGTHS Find the value of x. 5 4 x 2 2

FINDING SEGMENT LENGTHS Find the value of x. x 9 7

FINDING SEGMENT LENGTHS Find the value of x. 24 12

FINDING SEGMENT LENGTHS Find the value of x. 12 x 1 4

ERROR ANALYSIS Describe and correct the error in finding CD. CD p DF 5 AB p AF CD p 4 5 5 p 3 CD p 4 5 15 CD 5 3.75 A B F D C

FINDING SEGMENT LENGTHS Find the value of x. Round to the nearest tenth. 2x x 1 3 15 12

FINDING SEGMENT LENGTHS Find the value of x. Round to the nearest tenth. 45 27 50

FINDING SEGMENT LENGTHS Find the value of x. Round to the nearest tenth. 2 3

MULTIPLE CHOICE Which of the following is a possible value of x? A 22 B 4 C 5 D 6 2x 1 6 2

FINDING LENGTHS Find PQ. Round your answers to the nearest tenth. 6 12 N P M

FINDING LENGTHS Find PQ. Round your answers to the nearest tenth. 2 14 P P

CHALLENGE In the figure, AB 5 12, BC 5 8, DE 5 6, PD 5 4, and A is a point of tangency. Find the radius of (P. A P E D C

ARCHAEOLOGY The circular stone mound in Ireland called Newgrange has a diameter of 250 feet. A passage 62 feet long leads toward the center of the mound. Find the perpendicular distance x from the end of the passage to either side of the mound.

PROVING THEOREM 10.14 Write a two-column proof of Theorem 10.14. Use similar triangles as outlined in the Plan for Proof on page 689.

WELLS In the diagram of the water well, AB, AD, and DE are known. Write an equation for BC using these three measurements. G F B C E D A

PROOF Use Theorem 10.1 to prove Theorem 10.16 for the special case when the secant segment contains the center of the circle.

SHORT RESPONSE You are designing an animated logo for your website. Sparkles leave point C and move to the circle along the segments shown so that all of the sparkles reach the circle at the same time. Sparkles travel from point C to point D at 2 centimeters per second. How fast should sparkles move from point C to point N? Explain. . # CM CM 

PROVING THEOREM 10.15 Use the plan to prove Theorem 10.15. GIVEN c }EB and }ED are secant segments. PROVE c EA p EB 5 EC p ED Plan for Proof Draw }AD and }BC . Show that nBCE and nDAE are similar. Use the fact that corresponding side lengths in similar triangles are proportional. A C B D

PROVING THEOREM 10.16 Use the plan to prove Theorem 10.16. GIVEN c }EA is a tangent segment. }ED is a secant segment. PROVE c EA2 5 EC p ED Plan for Proof Draw }AD and }AC. Use the fact that corresponding side lengths in similar triangles are proportional. D C A E

EXTENDED RESPONSE In the diagram, }EF is a tangent segment, mADC 5 1408, mCAB 5 208, m EFD 5 608, AC 5 6, AB 5 3, and DC 5 10. a. Find m CAB. b. Show that nABC , nFEC. c. Let EF 5 y and DF 5 x. Use the results of part (b) to write a proportion involving x and y. Solve for y. d. Use a theorem from this section to write another equation involving both x and y. e. Use the results of parts (c) and (d) to solve for x and y. f. Explain how to find CE. 0 6 3 C E F D A

CHALLENGE Stereographic projection is a map-making technique that takes points on a sphere with radius one unit (Earth) to points on a plane (the map). The plane is tangent to the sphere at the origin. The map location for each point P on the sphere is found by extending the line that connects N and P. The points projection is where the line intersects the plane. Find the distance d from the point P to its corresponding point P9(4, 23) on the plane. Equator P(4, 3) P d y x N

Evaluate the expression. (p. 874) }} (210)2 2 82

Evaluate the expression. (p. 874) }} 25 1 (24) 1 (6 2 1)2

Evaluate the expression. (p. 874) }}} [22 2 (26)]2 1 (3 2 6)2

In right nPQR, PQ 5 8, m Q 5 408, and m R 5 508. Find QR and PR to the nearest tenth. (p. 473)

EF is tangent to (C at E. The radius of (C is 5 and EF 5 8. Find FC. (p. 651)

Find the indicated measure. }AC and}BE are diameters. (p. 659) A E C D B 1358 608 mCAB

Find the indicated measure. }AC and}BE are diameters. (p. 659) A E C D B 1358 608 mCCD

Find the indicated measure. }AC and}BE are diameters. (p. 659) A E C D B 1358 608 mCBCA

Find the indicated measure. }AC and}BE are diameters. (p. 659) A E C D B 1358 608 mCCBD

Find the indicated measure. }AC and}BE are diameters. (p. 659) A E C D B 1358 608 mCCDA

Find the indicated measure. }AC and}BE are diameters. (p. 659) A E C D B 1358 608 mCBAE

Determine whether}AB is a diameter of the circle. Explain. (p. 664) S B A 6 7

Determine whether}AB is a diameter of the circle. Explain. (p. 664) A B C D 10

Determine whether}AB is a diameter of the circle. Explain. (p. 664) D 3.2 4 4 5

VOCABULARY Copy and complete: The standard equation of a circle can be written for any circle with known ? and ? .

WRITING Explain why the location of the center and one point on a circle is enough information to draw the rest of the circle.

WRITING EQUATIONS Write the standard equation of the circle. y 1 1

WRITING EQUATIONS Write the standard equation of the circle. y 1 1

WRITING EQUATIONS Write the standard equation of the circle. y 10 10

WRITING EQUATIONS Write the standard equation of the circle. y 15 5

WRITING EQUATIONS Write the standard equation of the circle. y 10 10

WRITING EQUATIONS Write the standard equation of the circle. y 3 3

WRITING EQUATIONS Write the standard equation of the circle with the given center and radius. Center (0, 0), radius 7

WRITING EQUATIONS Write the standard equation of the circle with the given center and radius. Center (24, 1), radius 1

WRITING EQUATIONS Write the standard equation of the circle with the given center and radius. Center (7, 26), radius 8

WRITING EQUATIONS Write the standard equation of the circle with the given center and radius. Center (4, 1), radius 5

WRITING EQUATIONS Write the standard equation of the circle with the given center and radius. Center (3, 25), radius 7

WRITING EQUATIONS Write the standard equation of the circle with the given center and radius. Center (23, 4), radius 5

ERROR ANALYSIS Describe and correct the error in writing the equation of a circle. An equation of a circle with center (23, 25) and radius 3 is (x 2 3)2 1 (y 2 5)2 5 9.

MULTIPLE CHOICE The standard equation of a circle is (x 2 2)2 1 (y 1 1)2 5 16. What is the diameter of the circle? A 2 B 4 C 8 D 16

WRITING EQUATIONS Use the given information to write the standard equation of the circle. The center is (0, 0), and a point on the circle is (0, 6).

WRITING EQUATIONS Use the given information to write the standard equation of the circle. The center is (1, 2), and a point on the circle is (4, 2).

WRITING EQUATIONS Use the given information to write the standard equation of the circle. The center is (23, 5), and a point on the circle is (1, 8).

GRAPHING CIRCLES Graph the equation. x2 1 y 2 5 49

GRAPHING CIRCLES Graph the equation. (x 2 3)2 1 y 2 5 16

GRAPHING CIRCLES Graph the equation. x2 1 (y 1 2)2 5 36

GRAPHING CIRCLES Graph the equation. (x 2 4)2 1 (y 2 1)2 5 1

GRAPHING CIRCLES Graph the equation. (x 1 5)2 1 (y 2 3)2 5 9

GRAPHING CIRCLES Graph the equation. (x 1 2)2 1 (y 1 6)2 5 25

MULTIPLE CHOICE Which of the points does not lie on the circle described by the equation (x 1 2)2 1 (y 2 4)2 5 25? A (22, 21) B (1, 8) C (3, 4) D (0, 5)

ALGEBRA Determine whether the given equation defines a circle. If the equation defines a circle, rewrite the equation in standard form. x2 1 y 2 2 6y 1 9 5 4

ALGEBRA Determine whether the given equation defines a circle. If the equation defines a circle, rewrite the equation in standard form. x2 2 8x 1 16 1 y 2 1 2y 1 4 5 25

ALGEBRA Determine whether the given equation defines a circle. If the equation defines a circle, rewrite the equation in standard form. . x2 1 y 2 1 4y 1 3 5 16

ALGEBRA Determine whether the given equation defines a circle. If the equation defines a circle, rewrite the equation in standard form. x2 2 2x 1 5 1 y 2 5 81

IDENTIFYING TYPES OF LINES Use the given equations of a circle and a line to determine whether the line is a tangent, secant, secant that contains a diameter, or none of these. Circle: (x 2 4)2 1 (y 2 3)2 5 9 Line: y 5 23x 1 6

IDENTIFYING TYPES OF LINES Use the given equations of a circle and a line to determine whether the line is a tangent, secant, secant that contains a diameter, or none of these. Circle: (x 1 2)2 1 (y 2 2)2 5 16 Line: y 5 2x 2 4

IDENTIFYING TYPES OF LINES Use the given equations of a circle and a line to determine whether the line is a tangent, secant, secant that contains a diameter, or none of these. Circle: (x 2 5)2 1 (y 1 1)2 5 4 Line: y 5 1 }5 x 2 3

IDENTIFYING TYPES OF LINES Use the given equations of a circle and a line to determine whether the line is a tangent, secant, secant that contains a diameter, or none of these. Circle: (x 1 3)2 1 (y 2 6)2 5 25 Line: y 5 24 }3x 1 2

CHALLENGE Four tangent circles are centered on the x-axis. The radius of (A is twice the radius of (O. The radius of (B is three times the radius of (O. The radius of (C is four times the radius of (O. All circles have integer radii and the point (63, 16) is on (C. What is the equation of (A?

COMMUTER TRAINS A citys commuter system has three zones covering the regions described. Zone 1 covers people living within three miles of the city center. Zone 2 covers those between three and seven miles from the center, and Zone 3 covers those over seven miles from the center. a. Graph this situation with the city center at the origin, where units are measured in miles. b. Find which zone covers people living at (3, 4), (6, 5), (1, 2), (0, 3), and (1, 6). Zone 1 Zone 2 Z

COMPACT DISCS The diameter of a CD is about 4.8 inches. The diameter of the hole in the center is about 0.6 inches. You place a CD on the coordinate plane with center at (0, 0). Write the equations for the outside edge of the disc and the edge of the hole in the center. 4.8 in. 0.6 in.

REULEAUX POLYGONS In Exercises 3841, use the following information. The figure at the right is called a Reuleaux polygon. It is not a true polygon because its sides are not straight. nABC is equilateral. E F D G J H JD lies on a circle with center A and radius AD. Write an equation of this circle.

REULEAUX POLYGONS In Exercises 3841, use the following information. The figure at the right is called a Reuleaux polygon. It is not a true polygon because its sides are not straight. nABC is equilateral. E F D G J H DEC lies on a circle with center B and radius BD. Write an equation of this circle.

REULEAUX POLYGONS In Exercises 3841, use the following information. The figure at the right is called a Reuleaux polygon. It is not a true polygon because its sides are not straight. nABC is equilateral. E F D G J H CONSTRUCTION The remaining arcs of the polygon are constructed in the same way asCJD andCDE in Exercises 38 and 39. Construct a Reuleaux polygon on a piece of cardboard.

REULEAUX POLYGONS In Exercises 3841, use the following information. The figure at the right is called a Reuleaux polygon. It is not a true polygon because its sides are not straight. nABC is equilateral. E F D G J H Cut out the Reuleaux polygon from Exercise 40. Roll it on its edge like a wheel and measure its height when it is in different orientations. Explain why a Reuleaux polygon is said to have constant width.

EXTENDED RESPONSE Telecommunication towers can be used to transmit cellular phone calls. Towers have a range of about 3 km. A graph with units measured in kilometers shows towers at points (0, 0), (0, 5), and (6, 3). a. Draw the graph and locate the towers. Are there any areas that may receive calls from more than one tower? b. Suppose your home is located at (2, 6) and your school is at (2.5, 3). Can you use your cell phone at either or both of these locations? c. City A is located at (22, 2.5) and City B is at (5, 4). Each city has a radius of 1.5 km. Which city seems to have better cell phone coverage? Explain.

REASONING The lines y 5 3 }4 x 1 2 and y 5 23 }4x 1 16 are tangent to (C at the points (4, 5) and (4, 13), respectively. a. Find the coordinates of C and the radius of (C. Explain your steps. b. Write the standard equation of (C and draw its graph.

PROOF Write a proof. GIVEN c A circle passing through the points (21, 0) and (1, 0) PROVE c The equation of the circle is x2 2 2yk 1 y 2 5 1 with center at (0, k). y (21, 0) (1, 0)

CHALLENGE The intersecting lines m and n are tangent to (C at the points (8, 6) and (10, 8), respectively. a. What is the intersection point of m and n if the radius r of (C is 2? What is their intersection point if r is 10? What do you notice about the two intersection points and the center C? b. Write the equation that describes the locus of intersection points of m and n for all possible values of r.

Find the perimeter of the figure. (p. 49) 22 in. 9 in.

Find the perimeter of the figure. (p. 49) 18 ft

Find the perimeter of the figure. (p. 433) 40 m 57 m

Find the circumference of the circle with given radius r or diameter d. Use p 5 3.14. (p. 49) r 5 7 cm

Find the circumference of the circle with given radius r or diameter d. Use p 5 3.14. (p. 49) d 5 160 in.

Find the circumference of the circle with given radius r or diameter d. Use p 5 3.14. (p. 49) d 5 48 yd

Find the radius r of (C. (p. 651) 15 C

Find the radius r of (C. (p. 651) 15 r r 21 C

Find the radius r of (C. (p. 651) 28 20

Copy and complete: If a chord passes through the center of a circle, then it is called a(n) ? .

Draw and describe an inscribed angle and an intercepted arc.

WRITING Describe how the measure of a central angle of a circle relates to the measure of the minor arc and the measure of the major arc created by the angle.

In Exercises 46, match the term with the appropriate segment. A. }LM B. }KL C. }LN N K M L Tangent segment

In Exercises 46, match the term with the appropriate segment. A. }LM B. }KL C. }LN N K M L Secant segment

In Exercises 46, match the term with the appropriate segment. A. }LM B. }KL C. }LN N K M L External segment

Find the value of the variable. Y and Z are points of tangency on (W Z W 9a2 2 30 3a

Find the value of the variable. Y and Z are points of tangency on (W X Y Z W 2c2 1 9c 1 6 9c 1 14

Find the value of the variable. Y and Z are points of tangency on (W X W Z r 3 r

Use the diagram above to find the measure of the indicated arc. KL

Use the diagram above to find the measure of the indicated arc. LM

Use the diagram above to find the measure of the indicated arc. KM

Use the diagram above to find the measure of the indicated arc. KN

Find the measure ofCAB . E A D B C 618

Find the measure ofCAB . B E 658

Find the measure ofCAB . A E C 918

Find the value(s) of the variable(s). X Y Z 568 c8

Find the value(s) of the variable(s). C A 40

Find the value(s) of the variable(s). E F D G 1008 4r 8 808

Find the value of x. 2508 x8

Find the value of x. 8 968 4

Find the value of x. 1528 x8 608

SKATING RINK A local park has a circular ice skating rink. You are standing at point A, about 12 feet from the edge of the rink. The distance from you to a point of tangency on the rink is about 20 feet. Estimate the radius of the rink. D B 12 ft A r r 20 ft

Write an equation of the circle shown. y 1 2

Write an equation of the circle shown. y 2 2

Write an equation of the circle shown. y 2 2

Write the standard equation of the circle with the given center and radius. Center (0, 0), radius 9

Write the standard equation of the circle with the given center and radius. Center (25, 2), radius 1.3

Write the standard equation of the circle with the given center and radius. Center (6, 21), radius 4

Write the standard equation of the circle with the given center and radius. Center (23, 2), radius 16

Write the standard equation of the circle with the given center and radius. Center (10, 7), radius 3.5

Write the standard equation of the circle with the given center and radius. Center (0, 0), radius 5.2