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Topic 12 TOPIC OVERVIEW VOCABULARY 12-1Tangent Lines 12-2Chords and Arcs 12-3Inscribed Angles 12-4Angle Measures and Segment Lengths DIGITAL Theorems About Circles APPS English/Spanish Vocabulary Audio Online: EnglishSpanish chord, p. 492cuerda inscribed angle, p. 499 ángulo inscrito intercepted arc, p. 499 arco interceptor point of tangency, p. 486 punto de tangencia secant, p. 505secante tangent to a circle, p. 486 tangente de un círculo PRINT and eBook Access Your Homework . . . Online homework You can do all of your homework online with built-in examples and “Show Me How” support! When you log in to your account, you’ll see the homework your teacher has assigned you. Your Digital Resources PearsonTEXAS.com Homework Tutor app Do your homework anywhere! You can access the Practice and Application Exercises, as well as Virtual Nerd tutorials, with this Homework Tutor app, available on any mobile device. STUDENT TEXT AND Homework Helper Access the Practice and Application Exercises that you are assigned for homework in the Student Text and Homework Helper, which is also available as an electronic book. 484 Topic 12 Theorems About Circles 3--Act Math Earth Watch Scientists estimate that there are currently about 3000 operational man-made satellites orbiting the Earth. These satellites serve different purposes, from communication to navigation and global positioning. Some are weather satellites that collect environmental information. The International Space Station is the largest man-made satellite that orbits the Earth. It serves as a space environment research facility, and it also offers astronauts amazing views of the Earth. Think about this as you watch this 3-Act Math video. Scan page to see a video for this 3-Act Math Task. If You Need Help . . . Vocabulary Online You’ll find definitions of math terms in both English and Spanish. All of the terms have audio support. Learning Animations You can also access all of the stepped-out learning animations that you studied in class. Interactive Math tools These interactive math tools give you opportunities to explore in greater depth key concepts to help build understanding. Interactive exploration You’ll have access to a robust assortment of interactive explorations, including interactive concept explorations, dynamic activitites, and topiclevel exploration activities. Student Companion Refer to your notes and solutions in your Student Companion. Remember that your Student Companion is also available as an ACTIVebook accessible on any digital device. Virtual Nerd Not sure how to do some of the practice exercises? Check out the Virtual Nerd videos for stepped-out, multi-level instructional support. PearsonTEXAS.com 485 12-1 Tangent Lines TEKS FOCUS VOCABULARY •Point of tangency – the point where a circle and a tangent intersect •Tangent to a circle – a line in the plane of the circle that intersects TEKS (12)(A) Apply theorems about circles, including relationships among angles, radii, chords, tangents, and secants, to solve noncontextual problems. the circle in exactly one point •Analyze – closely examine objects, ideas, or relationships to learn TEKS (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas. more about their nature Additional TEKS (1)(G), (6)(A), (9)(B) ESSENTIAL UNDERSTANDING A radius of a circle and the tangent that intersects the endpoint of the radius on the circle have a special relationship. Key Concept Tangent Lines A B A tangent to a circle is a line in the plane of the circle that intersects the circle in exactly one point. The point where a circle and a tangent intersect is the point of tangency. 3 BA is a tangent ray, and BA is a tangent segment. Theorem 12-1 Theorem If a line is tangent to a circle, then the line is perpendicular to the radius at the point of tangency. If . . . < > AB is tangent to } O at P Then . . . < > AB # OP A A P P O B O B For a proof of Theorem 12-1, see the Reference section on page 683. hsm11gmse_1201_t05166 486 Lesson 12-1 Tangent Lines hsm11gmse_1201_t05174 Theorem 12-2 Theorem If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle. If . . . < > AB # OP at P Then . . . < > AB is tangent to } O A P O B You will prove Theorem 12-2 in Exercise 19. Theorem 12-3 hsm11gmse_1201_t05174 Theorem If two tangent segments to a circle share a common endpoint outside the circle, then the two segments are congruent. If . . . Then . . . BA and BC are tangent to } O BA ≅ BC A B O C You will prove Theorem 12-3 in Exercise 12. hsm11gmse_1201_t05177 Problem 1 TEKS Process Standard (1)(F) Finding Angle Measures Multiple Choice ML and MN are tangent to } O. What is the value of x? L O 117 x M N 58 What kind of angle is formed by a radius and a tangent? The angle formed is a right angle, so the measure is 90. 63 90 117 Since ML and MN are tangent to } O, ∠L and ∠N are right angles (Theorem 12-1). LMNO is a quadrilateral. Sohsm11gmse_1201_t05167 the sum of the angle measures is 360. m∠L + m∠M + m∠N + m∠O = 360 90 + m∠M + 90 + 117 = 360 297 + m∠M = 360 m∠M = 63 Substitute. Simplify. Solve. The correct answer is B. PearsonTEXAS.com 487 Problem 2 Finding Distance How does knowing Earth’s radius help? The radius forms a right angle with a tangent line from the observation deck to the horizon. So you can use two radii, the tower’s height, and the tangent to form a right triangle. STEM Earth Science The CN Tower in Toronto, Canada, has an observation deck 447 m above ground level. About how far is it from the observation deck to the horizon? Earth’s radius is about 6400 km. Step 1 Make a sketch. The length 447 m is about 0.45 km. T 447 m 0.45 km E C 6400 km Not to scale Step 2 Use the Pythagorean Theorem. CT 2 = TE 2 + CE 2 hsm11gmse_1201_t07766 (6400 + 0.45)2 = TE 2 + 64002 (6400.45)2 = TE 2 + 64002 Substitute. Simplify. 40,965,760.2025 = TE 2 + 40,960,000 5760.2025 = TE 2 76 ≈ TE Use a calculator. Subtract 40,960,000 from each side. Take the positive square root of each side. The distance from the CN Tower to the horizon is about 76 km. Problem 3 Finding a Radius B What is the radius of }C? Why does the value x appear on each side of the equation? The length of AC, the hypotenuse, is the radius plus 8, which is on the left side of the equation. On the right side of the equation, the radius is one side of the triangle. 488 AC 2 = AB2 + BC 2 (x + 8)2 = 122 + x2 Pythagorean Theorem Substitute. x2 + 16x + 64 = 144 + x2 16x = 80 Subtract x 2 and 64 from each side. x=5 Divide each side by 16. The radius is 5. Lesson 12-1 Tangent Lines Simplify. 12 x C A 8 x Problem 4 TEKS Process Standard (1)(G) Identifying a Tangent Is ML tangent to } N at L? Explain. What information does the diagram give you? •LMN is a triangle. •NM = 25, LM = 24, NL = 7 •NL is a radius. 25 N 7 The lengths of the sides of △LMN M 24 L To determine ML is a tangent if ML # NL. Use the whether ML is Converse of the Pythagorean Theorem to hsm11gmse_1201_t05175 tangent to } N determine whether △LMN is a right triangle. NL2 + ML2 ≟ NM 2 72 + 242 ≟ 252 625 = 625 Substitute. Simplify. By the Converse of the Pythagorean Theorem, △LMN is a right triangle with ML # NL. So ML is tangent to }N at L because it is perpendicular to the radius at the point of tangency (Theorem 12-2). Problem 5 Circles Inscribed in Polygons } O is inscribed in △ABC. What is the perimeter of △ABC? How can you find the length of BC? Find the segments congruent to BE and EC. Then use segment addition. A 10 cm D 15 cm B O F E 8 cm C AD = AF = 10 cm BD = BE = 15 cm CF = CE = 8 cm Thm 12-3: Two segments tangent to a circle from a point outside the circle are congruent, so they have the same length. hsm11gmse_1201_t05180 P = AB + BC + CA Definition of perimeter = AD + DB + BE + EC + CF + FA Segment Addition Postulate = 10 + 15 + 15 + 8 + 8 + 10 Substitute. = 66 The perimeter is 66 cm. PearsonTEXAS.com 489 HO ME RK O NLINE WO PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd™ tutorial video. Lines that appear to be tangent are tangent. O is the center of each circle. What is the value of x? For additional support when completing your homework, go to PearsonTEXAS.com. 1. 2. 3. O x x O 43 60 STEM 60 O x h Apply Mathematics (1)(A) The circle at the right represents Earth. The radius of d Earth is about 6400 km. Find the distance d to the horizon that a person can see r on a clear day from each of the following heights h above Earth. Round your answer hsm11gmse_1201_t05183 hsm11gmse_1201_t05182 hsm11gmse_1201_t05184 E r to the nearest tenth of a kilometer. 4. 5 km 5.1 km 6.2500 m Analyze Mathematical Relationships (1)(F) In each circle, what is the value of x, to the nearest tenth? 7. 8. x 14 x 10 cm 7 cm x x O x P 15 in. 9. O x hsm11gmse_1201_t05195 Q 9 in. 10 Each polygon circumscribes a circle. What is the perimeter of each polygon? 10. 16 cm 8 cm 1.9 in. 11. hsm11gmse_1201_t05192 hsm11gmse_1201_t05193 hsm11gmse_1201_t07772 3.7 in. 3.4 in. 6 cm 3.6 in. 9 cm A B 12.Write a paragraph proof to prove Theorem 12-3. Proof hsm11gmse_1201_t05190 Given: BA and BC are tangent to } O at A and C, respectively. O hsm11gmse_1201_t05189 Prove: BA ≅ BC C 13.a. A belt fits snugly around the two circular pulleys. CE is an auxiliary line from E to BD. CE } BA. What type of quadrilateral is ABCE ? Explain. b.What is the length of CE? B 14 in. C 35 in. A c.What is the distance between the centers of the pulleys to the nearest tenth? 490 Lesson 12-1 Tangent Lines 8 in. E hsm11gmse_1201_t05202 D hsm11gmse_1201_t07773 A 14.Explain Mathematical Ideas (1)(G) A nickel, a dime, and a quarter are touching as shown. Tangents are drawn from point A to both sides of each coin. What can you conclude about the four tangent segments? Explain. 15.Analyze Mathematical Relationships (1)(F) Leonardo da Vinci wrote, “When each of two squares touches the same circle at four points, one is double the other.” Explain why the statement is true. 16.Two circles that have one point in common are tangent circles. Given any triangle, explain how to draw three circles that are centered at each vertex of the triangle and tangent to each other. 17.Given: BC is tangent to }A at D. Proof DB ≅ DC 18.Given: }A and }B with common tangents DF and CE Proof Prove: AB ≅ AC Prove: △GDC ∼ △GFE D A A B D E G C B F C 19.Write an indirect proof of Theorem 12-2. A Proof Given: AB # OP at P. P hsm11gmse_1201_t05203 Prove: AB is tangent to } O. hsm11gmse_1201_t05204 O B TEXAS Test Practice hsm11gmse_1201_t05174 Lines in }O that appear to be tangent are tangent. What is the value of x? 20. 21. O 114 x O 56 x 22.The perimeter of an equilateral triangle is 90 in. What is its area to the nearest square inch? hsm11gmse_1201_t05218 hsm11gmse_1201_t05219 PearsonTEXAS.com 491 12-2 Chords and Arcs TEKS FOCUS VOCABULARY •Chord – a segment whose endpoints are TEKS (12)(A) Apply theorems about circles, including relationships among angles, radii, chords, tangents, and secants, to solve noncontextual problems. on a circle TEKS (1)(C) Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. •Number sense – the understanding of what numbers mean and how they are related Additional TEKS (1)(A), (1)(G), (5)(A), (5)(C), (6)(A), (9)(B) ESSENTIAL UNDERSTANDING You can use information about congruent parts of a circle (or congruent circles) to find information about other parts of the circle (or circles). Theorem 12-4 and Its Converse Theorem Within a circle or in congruent circles, congruent central angles have congruent arcs. Converse Within a circle or in congruent circles, congruent arcs have congruent central angles. B A C O D ¬ ¬ If ∠AOB ≅ ∠COD, then AB ≅ CD. ¬ ¬ If AB ≅ CD, then ∠AOB ≅ ∠COD. You will prove Theorem hsm11gmse_1202_t08234 12-4 and its converse in Exercises 7 and 24. Theorem 12-5 and Its Converse Theorem Within a circle or in congruent circles, congruent central angles have congruent chords. Converse Within a circle or in congruent circles, congruent chords have congruent central angles. B A C O D If ∠AOB ≅ ∠COD, then AB ≅ CD. If AB ≅ CD, then ∠AOB ≅ ∠COD. You will prove Theorem 12-5 and its converse in Exercises 8 and 25. hsm11gmse_1202_t08235 492 Lesson 12-2 Chords and Arcs Theorem 12-6 and Its Converse Theorem Within a circle or in congruent circles, congruent chords have congruent arcs. ¬ ¬ If AB ≅ CD, then AB ≅ CD . ¬ ¬ If AB ≅ CD, then AB ≅ CD. C B O Converse Within a circle or in congruent circles, congruent arcs have congruent chords. A D You will prove Theorem 12-6 and its converse in Exercises 9 and 26. hsm11gmse_1202_t08236 Theorem 12-7 and Its Converse Theorem Within a circle or in congruent circles, chords equidistant from the center or centers are congruent. BC E A Converse Within a circle or in congruent circles, congruent chords are equidistant from the center (or centers). O F If OE = OF, then AB ≅ CD. If AB ≅ CD, then OE = OF. D For a proof of Theorem 12-7, see the Reference section on page 683. You will prove the converse of Theorem 12-7 in Exercise 27. hsm11gmse_1202_t08242.ai Theorem 12-8 Theorem In a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc. If . . . AB is a diameter and AB # CD Then . . . ¬ ¬ CE ≅ ED and CA ≅ AD C C E A O B E A B O D D You will prove Theorem 12-8 in Exercise 10. Theorem 12-9 Theorem In a circle, if a diameter bisects a chord (that is not a diameter), then it is perpendicular to the chord. hsm11gmse_1202_t08238.ai hsm11gmse_1202_t08239.ai If . . . AB is a diameter and CE ≅ ED Then . . . AB # CD C C A E O D B A E O B D For a proof of Theorem 12-9, see the Reference section on page 683. hsm11gmse_1202_t08240.ai hsm11gmse_1202_t08241.ai PearsonTEXAS.com 493 Theorem 12-10 Theorem In a circle, the perpendicular bisector of a chord contains the center of the circle. If . . . AB is the perpendicular bisector of chord CD C A Then . . . AB contains the center of }O C A B B O D D You will prove Theorem 12-10 in Exercise 11. hsm11gmse_1202_t08244.ai Problem 1 hsm11gmse_1202_t08245.ai Using Congruent Chords Why is it important that the circles are congruent? The central angles will not be congruent unless the circles are congruent. B In the diagram, }O @ }P. Given that BC @ DF, what can you conclude? ∠O ≅ ∠P because, within congruent circles, congruent chords have ¬ ¬ congruent central angles (conv. of Thm. 12-5). BC ≅ DF because, within congruent circles, congruent chords have congruent arcs (Thm. 12-6). O D P C F hsm11gmse_1202_t06826.ai Problem 2 Finding the Length of a Chord What is the length of RS in }O? S O 9 The diagram indicates that PQ = QR = 12.5 and PR and RS are both 9 units from the center. P 9 Q R PR ≅ RS, since they are the same distance from the center of the circle. So finding PR gives the length of RS. The length of chord RS 494 12.5 hsm11gmse_1202_t06828 PQ = QR = 12.5 Given in the diagram PQ + QR = PR Segment Addition Postulate 12.5 + 12.5 = PR 25 = PR RS = PR Chords equidistant from the center of a circle are congruent (Theorem 12-7). RS = 25 Lesson 12-2 Chords and Arcs Substitute. Add. Substitute. 2 5 . . . . . . . 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 Problem 3 TEKS Process Standard (1)(C) Investigating Special Segments of Circles Choose from a variety of tools (such as a compass, straightedge, geometry software, and pencil and paper) to investigate the perpendicular bisectors of chords of a circle. Draw a circle with two chords that are not diameters. Construct the perpendicular bisectors of the chords. Then make a conjecture about the perpendicular bisector of a chord. You can construct perpendicular bisectors using paper, a pencil, a compass, and a straightedge. You can draw the circle and chords using a compass and straightedge, and construct the perpendicular bisectors using paper folding. Step 1 Use a compass to draw a circle on a piece of paper. Step 2Use a straightedge to draw two chords that are not diameters. Why should you draw more than two chords? The more examples you can find to support your conjecture, the stronger your conjecture becomes. Step 3Fold the perpendicular bisector for each chord. The perpendicular bisectors appear to intersect at the center of the circle. Step 4Draw a third chord and construct its perpendicular bisector. The third perpendicular bisector also appears to intersect the other two. Conjecture: The perpendicular bisector of any chord of a circle goes through the center of the circle. Problem 4 TEKS Process Standard (1)(A) Using Diameters and Chords How does the construction help find the center? The perpendicular bisectors contain diameters of the circle. Two diameters intersect at the circle’s center. Archaeology An archaeologist found pieces of a jar. She wants to find the radius of the rim of the jar to help guide her as she reassembles the pieces. What is the radius of the rim? Step 1 Trace a piece of the rim. Draw two chords and construct perpendicular bisectors. Step 2 The center is the intersection of the perpendicular bisectors. Use the center to find the radius. 0 1 2 3 4 5 6 inch The radius is 4 in. hsm11gmse_1202_t08246 hsm11gmse_1202_t08247 PearsonTEXAS.com 495 Problem 5 Finding Measures in a Circle Find two sides of a right triangle. The third side either is the answer or leads to an answer. Algebra What is the value of each variable to the nearest tenth? LN = 12(14) = 7 A diameter # to a chord bisects the chord (Theorem 12-8). A r L K 3 cm N r 2 = 32 + 72 M r ≈ 7.6 14 cm E B A Use the Pythagorean Theorem. Find the positive square root of each side. BC # AF A diameter that bisects a chord that is not a diameter is # to the chord (Theorem 12-9). 15 11 hsm11gmse_1202_t06831 B C y BA = BE = 15 Draw an auxiliary BA. The auxiliary BA ≅ BE because they are radii of the same circle. F y 2 + 112 = 152 y 2 = 104 11 Use the Pythagorean Theorem. Solve for y 2. y ≈ 10.2 Find the positive square root of each side. NLINE HO ME RK O hsm11gmse_1202_t06832 WO PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd™ tutorial video. In Exercises 1 and 2, the circles are congruent. What can you conclude? 1. B 2. For additional support when completing your homework, go to PearsonTEXAS.com. C A X Y E Z M T H F G L J K N Find the value of x. 3. 4. 5. hsm11gmse_1202_t06834 O 7 5 x 5 15 3.5 x x hsm11gmse_1202_t06835.ai 5 8 K G 6. Justify Mathematical Arguments (1)(G) In the diagram at the right, GH and KM are perpendicular bisectors of the chords they intersect. What can you conclude hsm11gmse_1202_t06841.ai hsm11gmse_1202_t06836.ai about the center of the circle? Justify your answer. hsm11gmse_1202_t06840.ai M H 496 Lesson 12-2 Chords and Arcs hsm11gmse_1202_t08251 7.Prove Theorem 12-4. Proof Given: } O with ∠AOB ≅ ∠COD ¬ ¬ Prove: AB ≅ CD B A Given: }O with ∠AOB ≅ ∠COD Prove: AB ≅ CD A C O 8.Prove Theorem 12-5. Proof O D 9.Prove Theorem 12-6. Proof B Proof hsm11gmse_1202_t08252 Given: / is the # bisector of WY. Prove: / contains the center of }X. D 10.Prove Theorem 12-8. Proof 11.Prove Theorem 12-10. C hsm11gmse_1202_t06853.ai 12.Given: }A with CE # BD Proof ¬ ¬ Prove: BC ≅ DC C X W Z A Y B E F D 13.a.Select Tools to Solve Problems (1)(C) Select a tool, such as compass and straightedge or geometry software, to construct two chords in a circle such that the chords are perpendicular and have a common endpoint. Draw the line hsm11gmse_1202_t06861.ai hsm11gmse_1202_t06863.ai segment that joins the other two endpoints. b.In the same circle, repeat part (a) several times, changing the lengths of the perpendicular chords each time. Then make a conjecture about the line segment that joins the endpoints of two perpendicular chords with a common endpoint. 14.Apply Mathematics (1)(A) You are building a circular patio table. You have to drill a hole through the center of the tabletop for an umbrella. How can you find the center? }A and }B are congruent. CD is a chord of both circles. 15.If AB = 8 in. and CD = 6 in., how long is a radius? 16.If AB = 24 cm and a radius = 13 cm, how long is CD? 17.If a radius = 13 ft and CD = 24 ft, how long is AB? C A B D 18.In the figure at the right, sphere O with radius 13 cm is intersected by a plane 5 cm from center O. Find the radius of the cross section }A. 19.A plane intersects a sphere that has radius 10 in., forming the cross section }B 5 cm O 13 cm with radius 8 in. How far is the plane from the center of the hsm11gmse_1202_t06858.ai sphere? 20.Connect Mathematical Ideas (1)(F) Two concentric circles have radii of 4 cm and 8 cm. A segment tangent to the smaller circle is a chord of the larger circle. What is the length of the segment to the nearest tenth? 21.Display Mathematical Ideas (1)(G) Use Theorem 12-5 to construct a regular octagon. A hsm11gmse_1202_t06852 PearsonTEXAS.com 497 22.In the diagram at the right, the endpoints of the chord are the points where the line x = 2 intersects the circle x2 + y 2 = 25. What is the length of the chord? Round your answer to the nearest tenth. 23.Explain Mathematical Ideas (1)(G) Theorems 12-4 and 12-5 both begin with the phrase “within a circle or in congruent circles.” Explain why the word congruent is essential for both theorems. y x2 3 x O 3 3 3 Prove each of the following. Proof 24.Converse of Theorem 12-4: Within a circle or in congruent circles, congruent arcs have congruent central angles. 25.Converse of Theorem 12-5: Within a circle or in congruent circles, congruent chords have congruent central angles. hsm11gmse_1202_t06859.ai 26.Converse of Theorem 12-6: Within a circle or in congruent circles, congruent arcs have congruent chords. 27.Converse of Theorem 12-7: Within a circle or congruent circles, congruent chords are equidistant from the center (or centers). 28.If two circles are concentric and a chord of the larger circle is tangent to the Proof smaller circle, prove that the point of tangency is the midpoint of the chord. C A 29.Analyze Mathematical Relationships (1)(F) In }O, AB is a diameter of the circle and AB # CD. What conclusions can you make? TEXAS Test Practice hsm11gmse_1202_t06842.ai A.9.0 cm C.18.0 cm B.9.6 cm D.19.2 cm 31.From the top of a building you look down at an object on the ground. Your eyes are 50 ft above the ground, and the angle of depression is 50°. Which distance is the best estimate of how far the object is from the base of the building? F. 42 ft H.65 ft G.60 ft J.78 ft 32.A bicycle tire has a diameter of 17 in. How many revolutions of the tire are necessary to travel 800 ft? Show your work. Lesson 12-2 Chords and Arcs B D 30.The diameter of a circle is 25 cm and a chord of the same circle is 16 cm. To the nearest tenth, what is the distance of the chord from the center of the circle? 498 E O 12-3 Inscribed Angles TEKS FOCUS VOCABULARY TEKS (12)(A) Apply theorems about circles, including relationships among angles, radii, chords, tangents, and secants, to solve noncontextual problems. TEKS (1)(G) Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication. •Inscribed angle – an angle whose vertex is on the circle and whose sides are chords of the circle •Justify – explain with logical reasoning. You can justify a mathematical argument. •Argument – a set of statements put forth to show the truth or falsehood of a mathematical claim Additional TEKS (1)(D), (5)(A) ESSENTIAL UNDERSTANDING Angles formed by intersecting lines have a special relationship to the arcs the intersecting lines intercept. In this lesson, you will study arcs formed by inscribed angles. Theorem 12-11 Inscribed Angle Theorem The measure of an inscribed angle is half the measure of its intercepted arc. ¬ m∠B = 12 mAC A B C You will prove Theorem 12-11 in Exericises 9 and 10. Corollaries to Theorem 12-11: hsm11gmse_1203_t06876 The Inscribed Angle Theorem Corollary 1 Two inscribed angles that intercept the same arc are congruent. Corollary 2 An angle inscribed in a semicircle is a right angle. Corollary 3 The opposite angles of a quadrilateral inscribed in a circle are supplementary. B A A B D C C You will prove these corollaries in Exercises 14–16. hsm11gmse_1203_t08254.ai hsm11gmse_1203_t08255.ai hsm11gmse_1203_t08256.ai PearsonTEXAS.com 499 Theorem 12-12 The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. B D B D C ¬ m∠C = 12 m BDC C You will prove Theorem 12-12 in Exercise 17. hsm11gmse_1203_t06899.ai Problem 1 Using the Inscribed Angle Theorem Which variable should you solve for first? You know the inscribed ¬ angle that intercepts PT , which has the measure a. You need a to find b. So find a first. What are the values of a and b? ¬ m∠PQT = 12 m PT Inscribed Angle Theorem Substitute. 60 = 12 a ¬ m∠PRS = 12 m PS ¬ ¬ m∠PRS = 12 (m PT + m TS ) 120 = a P a T Q 60 Multiply each side by 2. b 30 S R Inscribed Angle Theorem Arc Addition Postulate b = 12 (120 + 30) Substitute. b = 75 Simplify. hsm11gmse_1203_t06881 Problem 2 TEKS Process Standard (1)(D) Using Corollaries to Find Angle Measures What is the measure of each numbered angle? Is there too much information? Each diagram has more information than you need. Focus on what you need to find. A 40 1 500 70 B 70 ∠1 is inscribed in a semicircle. By Corollary 2, ∠1 is a right angle, so m∠1 = 90. hsm11gmse_1203_t06892 Lesson 12-3 Inscribed Angles 2 38 ∠2 and the 38° angle intercept the same arc. By Corollary 1, the angles are congruent, so m∠2 = 38. hsm11gmse_1203_t06893 Problem 3 TEKS Process Standard (1)(G) Using Arc Measure < > ¬ In the diagram, SR is tangent to the circle at Q. If mPMQ = 212, what is mjPQR? S Q < > M • SR is tangent to the circle at Q ¬ •m PMQ = 212 R P m∠PQS + m∠PQR = 180. So ¬ first find m∠PQS using PMQ . m∠PQR NLINE HO ME RK O How can you check the answer? One way is to use m∠PQR to find ¬ m PQ . Confirm that ¬ ¬ m PQ + m PMQ = 360. WO 1 ¬ 2 mPMQ = m∠PQS 1 2 (212) = m∠PQS 106 = m∠PQS The measure of an ∠ formed by a hsm11gmse_1203_t08257.ai tangent and a chord is 12 the measure of the intercepted arc. Substitute. Simplify. m∠PQS + m∠PQR = 180 Linear Pair Postulate 106 + m∠PQR = 180 Substitute. Simplify. m∠PQR = 74 PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd™ tutorial video. Find the value of each variable. For each circle, the dot represents the center. 95 1. 2. 3. 4. a For additional support when completing your homework, go to PearsonTEXAS.com. c 116 p 25 b b a c q 58 a Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Find the value of each variable. Lines that appear to be tangent are tangent. 246 5. 6. 7. hsm11gmse_1203_t06913 hsm11gmse_1203_t06915 hsm11gmse_1203_t06916 hsm11gmse_1203_t06906 y w x 230 115 e f 8. Justify Mathematical Arguments (1)(G) Can a rhombus that is not a square be inscribed in a circle? Justify your answer. hsm11gmse_1203_t06919.ai hsm11gmse_1203_t06918.ai hsm11gmse_1203_t06917.ai PearsonTEXAS.com 501 To prove Theorem 12-11, there are three cases to consider. In Case I, the center of the circle is on a side of the inscribed angle. In Case II, the center is inside the inscribed angle. In Case III, the center is outside the inscribed angle. Below is a proof of Case I. Proof Given: }O with inscribed j B and diameter BC ¬ Prove: mjB = 12 mAC C A O Draw radius OA to form isosceles △AOB with OA = OB. So B mjA = mjB by the Isosceles Triangle Theorem. Then, by the Triangle ¬ Exterior Angle Theorem, mjAOC = mjA + mjB. So mAC = mjAOC ¬ since j AOC is a central angle. This means that mAC = mjA + mjB, and ¬ ¬ therefore mAC = 2mjB. By the Division Property of Equality, mjB = 12 mAC . hsm11gmse_1203_t06880 Write a proof of Case II and Case III for Exercises 9 and 10. 9.Inscribed Angle Theorem, Case II Proof Given: }O with inscribed ∠ABC ¬ Prove: m∠ABC = 12 mAC 10.Inscribed Angle Theorem, Case III Proof Given: }S with inscribed ∠PQR ¬ Prove: m∠PQR = 12 m PR P R T C P A S O Q B 11.Explain Mathematical Ideas (1)(G) The director of a telecast wants the option of showing the same scene from three different views. hsm11gmse_1203_t06934.ai hsm11gmse_1203_t06935.ai a.Explain why cameras in the positions shown in the diagram will transmit the same scene. b.Will the scenes look the same when the director views them on the control room monitors? Explain. Scene Find the value of each variable. For each circle, the dot represents the center. 12. 13. c 44 b Lesson 12-3 Inscribed Angles hsm11gmse_1203_t06921.ai Camera 3 c a 160 e 502 Camera 2 d 120 b a Camera 1 56 hsm11gmse_1203_t06929.ai hsm11gmse_1203_t06922.ai Write a proof for Exercises 14–17. 14.Inscribed Angle Theorem, Corollary 1 15.Inscribed Angle Theorem, Corollary 2 Proof Proof ¬ Given: }O, ∠A intercepts BC , Given: }O with ∠CAB inscribed ¬ ∠D intercepts BC . in a semicircle Prove: ∠A ≅ ∠D Prove: ∠CAB is a right angle. B C A D O B O C D A 16.Inscribed Angle Theorem, Corollary 3 Proof 17.Theorem 12-12 Proof Given: Quadrilateral ABCD Given: GH and tangent / inscribed in }Ohsm11gmse_1203_t06938.ai intersecting }E at H hsm11gmse_1203_t06937.ai ¬ Prove: ∠A and ∠C are supplementary. Prove: m∠GHI = 12 m GFH ∠B and ∠D are supplementary. D A B F G I E O C H TEXAS Test Practice hsm11gmse_1203_t06939.ai hsm11gmse_1203_t06940.ai For Exercises 18 and 19, what is the value of each variable in }O? 18. A.25 O 130 x B.35 C.45 D.65 19. y O 60 F. 20 G.30 H.50 J.60 20.Is the following proof valid? If not, explain why, and then write a valid proof. B C hsm11gmse_1203_t06942.ai ∠ABC Given: Quadrilateral ABCD, ∠A ≅ ∠C, BD bisects hsm11gmse_1203_t06941.ai Prove: ∠ADB ≅ ∠CDB BD ≅ BD by the Reflexive Property. Since BD bisects ∠ABC, it also bisects ∠ADC. So ∠ADB ≅ ∠CDB. A D hsm11gmse_1203_t08261 503 PearsonTEXAS.com Technology Lab Use With Lesson 12-4 Exploring Chords and Secants teks (5)(A), (1)(E) 1 Construct }A and two chords BC and DE that intersect at F. E C 1.Measure BF, FC, EF, and FD. 2.Use the calculator program of your software to find BF # FC and EF # FD. F A B 3.Manipulate the lines. What pattern do you observe in the products? 4.What appears to be true for two intersecting chords? D 2 A secant is a line that intersects a circle in two points. A secant segment is a segment that contains a chord of the circle and has only one endpoint < > < > Ghsm11gmse_1204a_t06963.ai outside the circle. Construct a new circle and two secants DG and DE that intersect outside the circle at point D. Label the intersections with the circle as shown. F A 5.Measure DG, DF, DE, and DB. 6.Calculate the products DG # DF and DE # DB. D B E 7.Manipulate the lines. What pattern do you observe in the products? 8.What appears to be true for two intersecting secants? hsm11gmse_1204a_t06964.ai 3 Construct }A with tangent DG perpendicular to radius AG and secant DE that intersects the circle at B and E. 9.Measure DG, DE, and DB. 10. Calculate the products (DG)2 and DE G # DB. A 11. Manipulate the lines. What pattern do you observe in the products? 12. What appears to be true for the tangent segment and secant segment? D B E hsm11gmse_1204a_t06965.ai 504 Technology Lab Exploring Chords and Secants 12-4 Angle Measures and Segment Lengths TEKS FOCUS VOCABULARY TEKS (12)(A) Apply theorems about circles, including relationships among angles, radii, chords, tangents, and secants, to solve non-contextual problems. •Secant – a line that intersects a TEKS (1)(C) Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. •Number sense – the understanding circle at two points of what numbers mean and how they are related Additional TEKS (1)(D), (1)(F), (5)(A), (6)(A), (6)(D) ESSENTIAL UNDERSTANDING • Angles formed by intersecting lines have a special • There is a special relationship between two intersecting relationship to the related arcs formed when the lines intersect a circle. chords, two intersecting secants, or a secant that intersects a tangent. This relationship allows you to find the lengths of unknown segments. Theorem 12-13 The measure of an angle formed by two lines that intersect inside a circle is half the sum of the measures of the intercepted arcs. m∠1 = 12(x + y) x y 1 For a proof of Theorem 12-13, see the Reference section on page 683. Theorem 12-14 hsm11gmse_1204_t06968 .ai The measure of an angle formed by two lines that intersect outside a circle is half the difference of the measures of the intercepted arcs. x° y° 1 I: angle formed by two secants x° y° x° 1 II: angle formed by a secant and a tangent y° 1 III: angle formed by two tangents m∠1 = 12(x - y) You will prove Theorem 12-14 in Exercises 24 and 25. PearsonTEXAS.com 505 Theorem 12-15 For a given point and circle, the product of the lengths of the two segments from the point to the circle is constant along any line through the point and the circle. I. II. III. a d P a c x w b y t P P y z z # b = c # d (w + x)w = ( y + z)y ( y + z) y = t 2 You will prove Theorem 12-13 in Exercises 13 and 14. hsm11gmse_1204_t06982hsm11gmse_1204_t06983 hsm11gmse_1204_t06984 Problem 1 TEKS Process Standard (1)(C) Investigating Special Angles of Circles Choose from a variety of tools (such as a protractor, a compass, or geometry software) to investigate angles formed by chords intersecting inside a circle. Explain why you chose that tool. Construct a circle and several pairs of intersecting chords. Then make a conjecture about the measure of an angle formed by two chords intersecting inside a circle. Geometry software could help you investigate angles created by chords. You can create a circle and two chords and then drag the chords around the circle to investigate angle measures and make a conjecture. Which angles are formed by the intersection of the chords in the diagram? ∠CFD, ∠EFB, ∠CFE, and ∠DFB Step 1Draw a circle and two chords that intersect in the interior of the circle. Step 2Measure angles formed by the intersection of the chords and their intercepted arcs. Record your findings in the table below. E C F A Step 3Drag the chords to form different angles. Record your findings in a table. Vertical angle measure 65° 41° 81° Intercepted arc 44° 59° 30° Intercepted arc 86° 23° 132° 130° 82° 162° Sum of arc measures D hsm11gmse_1204a_t06963.ai Conjecture: The measure of an angle formed by two chords that intersect inside a circle is half the sum of the measures of its intercepted arcs. 506 B Lesson 12-4 Angle Measures and Segment Lengths Problem 2 Finding Angle Measures Algebra What is the value of each variable? A Remember to add arc measures for arcs intercepted by lines that intersect inside a circle and subtract arc measures for arcs intercepted by lines that intersect outside a circle. B 95 x 46 z 90 20 x = 12(46 + 90) Theorem 12-13 20 = 12(95 - z) Theorem 12-14 x = 68 Simplify. 40 = 95 - z hsm11gmse_1204_t06975.ai z = 55 Solve for z. hsm11gmse_1204_t06974.ai Multiply each side by 2. Problem 3 Finding an Arc Measure Satellite A satellite in a geostationary orbit above Earth’s equator has a viewing angle of Earth formed by the two tangents to the equator. The viewing angle is about 17.5°. What is the measure of the arc of Earth that is viewed from the satellite? A Earth E How can you represent the measures of the arcs? The sum of the measures of the arcs is 360°. If the measure of one arc is x, then the measure of the other is 360 - x. Satellite 17.5 B ¬ Let mAB = x. ¬ Then m AEB = 360 - x. hsm11gmse_1204_t08584.ai ¬ ¬ 17.5 = 12(m AEB - mAB ) Theorem 12-14 17.5 = 12[(360 - x) - x] Substitute. 17.5 = 12(360 - 2x) Simplify. 17.5 = 180 - x Distributive Property x = 162.5 Solve for x. A 162.5° arc can be viewed from the satellite. PearsonTEXAS.com 507 Problem 4 TEKS Process Standard (1)(F) Finding Segment Lengths How can you identify the segments needed to use Theorem 12-15? Find where segments intersect each other relative to the circle. The lengths of segments that are part of one line will be on the same side of an equation. Algebra Find the value of the variable in }N. A N 84 = 49 + 7y 35 = 7y 5=y HO RK O ME WO Thm. 12-15, Case II 8 16 (8 + 16)8 = z 2 Thm. 12-15, Case III 192 = z 2 Distributive Property hsm11gmse_1204_t06986 NLINE N (6 + 8)6 = (7 + y)7 z y 7 B 8 6 Simplify. hsm11gmse_1204_t06987 13.9 ≈ z Solve for z. Solve for y. PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd™ tutorial video. Find the value of each variable using the given chord, secant, and tangent lengths. If the answer is not a whole number, round to the nearest tenth. For additional support when completing your homework, go to PearsonTEXAS.com. 1. 15 26 2. x 20 3. 20 11 13 x y 5 7 15 c In the diagram at the right, CA and CB are tangents to }O. Write an expression for each arc or angle in terms of the given variable. ¬ ¬ 4. m ADB using x 5.m∠C using x 6.m AB using y A D O y x C hsm11gmse_1204_t06999 hsm11gmse_1204_t07002B hsm11gmse_1204_t07000 Find the value of each variable. 7. 8. 160 53 60 y x 68 x 9. y x hsm11gmse_1204_t07004 70 10.Analyze Mathematical Relationships (1)(F) You focus your camera on a circular fountain. Your camera is at the hsm11gmse_1204_t06997 vertex of the angle formed byhsm11gmse_1204_t06996 tangents to the fountain. hsm11gmse_1204_t06992 You estimate that this angle is 40°. What is the measure of the arc of the circular basin of the fountain that will be in the photograph? 508 Lesson 12-4 Angle Measures and Segment Lengths STEM 11.Apply Mathematics (1)(A) Arc The basis for the design x of the Wankel rotary engine 8 in. is an equilateral triangle. Each side of the triangle is a chord to an arc of a circle. The opposite vertex of the triangle is the center of the circle that forms the arc. In Center the diagram at the right, each side of the equilateral triangle is 8 in. long. Wankel engine a.Use what you know about equilateral triangles and find the value of x. hsm11gmse_1204_t06991 b.Justify Mathematical Arguments (1)(G) Copy the diagram and complete the circle with the given center. Then use Theorem 12-15 to find the value of x. Show that your answers to parts (a) and (b) are equal. 12.A circle is inscribed in a quadrilateral whose four angles have measures 85, 76, 94, and 105. Find the measures of the four arcs between consecutive points of tangency. Recall that there are three cases to prove Theorem 12-15. In Case I, the product of the chord segments are equal. In Case II, the products of the secants and their outer segments are equal. In Case III, the product of a secant and its outer segment equals the square of the tangent. Below is a proof of Case I. Proof Given: a circle with chords AB and CD intersecting at P Prove: a # b=c # d A Draw AC and BD. jA ≅ jD and jC ≅ jB because each pair intercepts the same arc, and angles that intercept the same arc are congruent. △APC = △DPB by the Angle-Angle Similarity Postulate. The lengths of a corresponding sides of similar triangles are proportional, so d = bc . Therefore, a b = c d. # C a d c P b B D # Use similar triangles to prove Case II and Case III. 13.Prove Theorem 12-15, Case II. Proof 14.Prove Theorem 12-15, Case III. Proof hsm11gmse_1204_t06985 15.The diagram at the right shows a unit circle, a circle with radius 1. E a.What triangle is similar to △ABE? b.Describe the connection between the tangent ratio for ∠A and the segment that is tangent to }A. D 1 A B C hypotenuse lengthoflegadjacenttoanangle . Describe the connection c.The secant ratio is between the secant ratio for ∠A and the segment that is the secant in }A. hsm11gmse_1204_t08590 .a PearsonTEXAS.com 509 For Exercises 16 and 17, use the diagram at the right. Prove each statement. ¬ ¬ 16.m∠1 + m PQ = 180 17.m∠1 + m∠2 = m QR Proof Q Proof 18.a. Select Tools to Solve Problems (1)(C) What tool(s) can you use to investigate the relationships between an angle formed by two perpendicular tangents to a circle and the arcs intercepted by the angle? Explain your choice. R 1 2 P b.Construct a circle with a several pairs of tangents that intersect at a right angle. Make a conjecture about the relationship between the right angle and hsm11gmse_1204_t07013 the minor arc intercepted by the angle. 19.In the diagram at the right, the circles are concentric. What is a formula you could use to find the value of c in terms of a and b? 20.△PQR is inscribed in a circle with m∠P = 70, m∠Q = 50, and ¬ ¬ ¬ m∠R = 60. What are the measures of PQ , QR , and PR ? c b a Find the values of x and y using the given chord, secant, and tangent lengths. If your answer is not a whole number, round it to the nearest tenth. 21. y 6 10 x 22. 8 x 4 16 23. y 24.Prove Case II of Theorem 12-14. 12 y 5 hsm11gmse_1204_t08588 .ai x A B Given: }O with secants CA and CE O C ¬ hsm11gmse_1204_t07009 ¬ 1 hsm11gmse_1204_t07010 D Prove: m∠ACE = 2 (m AE - m BD ) hsm11gmse_1204_t07008 E b P 25.Prove Case I and Case III of Theorem 12-14. Proof Proof 26.Use the diagram at the right and the theorems of this lesson to prove the Proof Pythagorean Theorem. a a Q R c O hsm11gmse_1204_t07012 a 27.If an equilateral triangle is inscribed in a circle, prove that the tangents Proof S to the circle at the vertices form an equilateral triangle. TEXAS Test Practice For Exercises 28 and 29, use the diagram at the right. hsm11gmse_1204_t07014 A 28.If BC = 6, DC = 5, and CE = 12, find AC. ¬ ¬ 29.If m∠C = 14 and m AE = 140, find m BD . B C E D hsm11gmse_1204_t07015 510 Lesson 12-4 Angle Measures and Segment Lengths Topic 12 Review TOPIC VOCABULARY • chord, p. 492 • point of tangency, p. 486 • inscribed angle, p. 499 • secant, p. 505 • tangent to a circle, p. 486 Check Your Understanding Use the figure to choose the correct term to complete each sentence. A D C X F E B < > 1. EF is (a secant of, tangent to) }X . 2.DF is a (chord, secant) of }X . hsm11gmse_12cr_t09116.ai 3.△ABC is made of (chords in, tangents to) }X . 4.∠DEF is an (intercepted arc, inscribed angle) of }X . 12-1 Tangent Lines Quick Review Exercises A tangent to a circle is a line that intersects the circle at exactly one point. The radius to that point is perpendicular to the tangent. From any point outside a circle, you can draw two segments tangent to a circle. Those segments are congruent. Use }O for Exercises 5–7. 5 A O 2 Example > B x > PA and PB are tangents. Find x. The radii are perpendicular to the tangents. Add the angle measures of the quadrilateral: x + 90 + 90 + 40 = 360 x + 220 = 360 x = 140 A O x 40 B P 60 3 C 5.What is the perimeter of △ABC? 6.OB = 128. What is the radius? hsm11gmse_12cr_t09053.ai 7. What is the value of x? hsm11gmse_12cr_t09048.ai PearsonTEXAS.com 511 12-2 Chords and Arcs Quick Review Exercises A chord is a segment whose endpoints are on a circle. Congruent chords are equidistant from the center. A diameter that bisects a chord that is not a diameter is perpendicular to the chord. The perpendicular bisector of a chord contains the center of the circle. Use the figure at the right for Exercises 8–10. Example What is the value of d? 8.If AB is a diameter and CE = ED, then m∠AEC = ? . 9.If AB is a diameter and is at right angles to CD, what is the ratio of CD to DE? + 122 = C D E 10. If CE = 12 CD and m∠DEB = 90, what is true of AB? hsm11gmse_12cr_t09054.ai B Use the circle below for Exercises 11 and 12. Since the chord is bisected, m∠ACB = 90. The radius is 13 units. So an auxiliary segment from A to B is 13 units. Use the Pythagorean Theorem. d2 A 9 13 B 12 132 d C 12 x hsm11gmse_12cr_t09056.ai 5 y 5 A d 2 = 25 11. What is the value of x? d=5 12. What is the value of y? hsm11gmse_12cr_t09055.ai hsm11gmse_12cr_t07095.ai 12-3 Inscribed Angles Quick Review Exercises Find the value of each variable. Line O is a tangent. An inscribed angle has its Intercepted A arc vertex on a circle, and its sides 13. 14. C B b are chords. An intercepted arc 40 Inscribed angle has its endpoints on the sides a a of an inscribed angle, and its d 20 c other points in the interior of the c b angle. The measure of an inscribed angle is half the measure of its hsm11gmse_1203_t06873 intercepted arc. b 15. c 16. ¬ What is m PS ? What is mjR? ¬ m∠Q = 60 is half m PS , so ¬ m PS = 120. ∠R intercepts the same arc as ∠Q, so m∠R = 60. a 59 b hsm11gmse_12cr_t09058.ai 72 Example P S d a hsm11gmse_12cr_t07097.ai 140 45 d c 60 Q R hsm11gmse_12cr_t07098.ai hsm11gmse_12cr_t07099.ai 512 Topic 12 Review hsm11gmse_12cr_t09057.ai 12-4 Angle Measures and Segment Lengths Quick Review Exercises A secant is a line that intersects a circle at two points. The following relationships are true: Find the value of each variable. c a x O d 1 17. x y 26 100 b abcd m1 1 (x y) 18. 145 2 hsm11gmse_12cr_t07106.ai a b x w O b B ahsm11gmse_12cr_t07102.ai z y 45 6 19. (w x)w ( y z)y 5 x mB 1 (a b) hsm11gmse_12cr_t07105.ai 2 10 a t O b hsm11gmse_12cr_t07103.ai z B y 20. ( y z)y t 2 5 10 hsm11gmse_12cr_t09059.ai x 8 mB 1 (a b) 2 Example What is the value of x? hsm11gmse_12cr_t07104.ai 19 (x + 10)10 = (19 + 9)9 9 10x + 100 = 252 x 10 x = 15.2 hsm11gmse_12cr_t07109.ai hsm11gmse_12cr_t07108.ai PearsonTEXAS.com 513 Topic 12 TEKS Cumulative Practice Multiple Choice 4.What is the value of x in the figure shown below? Read each question. Then write the letter of the correct answer on your paper. 1.A vertical mast is on top of building and is positioned 6 ft from the front edge. The mast casts a shadow perpendicular to the front of the building, and the tip of the shadow is 90 ft from the front of the building. At the same time, the 24-ft building casts a 64-ft shadow. What is the height of the mast? B. 9 ft 9 in.D. 33 ft 2.Javier leans a 20-ft-long ladder against a wall. If the base of the ladder is positioned 5 feet from the wall, how high up the wall does the ladder reach? Round to the nearest tenth. 3.△FGH has the vertices shown below. If the triangle is rotated 90° counterclockwise about the origin, what are the coordinates of the rotated point F′? y F x 4 2 4 5.A photographic negative is 3 cm by 2 cm. A similar print from the geom12_gm_c12_csr_t0005.ai negative is 9 cm long on its shorter side. What is the length of the longer side? A. 1.5 cmC. 12 cm B. 6 cmD. 13.5 cm G. 60° H. 75° 45 x 75 J. 105° 7.One side of a triangle has length 6 in. and another side has length 3 in. Which is the greatest possible whole-number value for the length of the third side? 4 2 F. 112H. 102 45° F. G. 18.8 ft J. 15 ft 4 2 76 6.What is the value of x? F. 19.4 ftH. 17.5 ft G x G. 104J. 89 A. 7 ft 6 in.C. 12 ft 102 H A. 3 in.C. 8 in. geom12_gm_c12_csr_t0004.ai B. 6 in.D. 9 in. ¬ 8.What is m RT in the figure at the right? F. 162° G. 146° ( -1, 2) A. (1, -3)C. B. (3, -1)D. (1, -2) T R 55 H. 110° J. 73° 104 S geom12_gm_c12_csr_t0002.ai 514 Topic 12 TEKS Cumulative Practice geom12_gm_c12_csr_t0003.ai Gridded Response Constructed Response 9.A ski ramp on a lake has the dimensions shown below. To the nearest hundredth of a meter, what is the height h of the ramp? 15. △DEF has vertices D(1, 1), E( -2, 4), and F(4, 7). What is the perimeter of △DEF ? Show your work. h 4.8 m 75 16. In }A below, AE = 13.1 and AC # BD. If BC = 6.8, what is AC, to the nearest tenth? Show your work. B D A 10. What is the value of n in the trapezoid shown below? C (23n) E geom12_gm_c12_csr_t0008.ai (14n5) 11. In the figure shown below, MN } OP, LM = 12, MN = 15, and MO = 6. What is OP? L geom12_gm_c12_csr_t0009.ai M 18. The endpoints of HK are H( -3, 7) and K(6, 1), and the endpoints of MN are M( -5, -8) and N(7, 10). What are the slopes of the line segments? Are the line segments parallel, perpendicular, or neither? Explaingeom12_gm_c12_csr_t0011.ai how you know and show your work. 19. Suppose a square is inscribed in a circle such as square HIJK shown below. N O 17. What is the measure of an exterior angle of a regular dodecagon (12-sided polygon)? Show your work. P 12. In the figure below, XY and XZ are tangent to }O at points Y and Z, respectively. What is m∠YOZ in degrees? Y geom12_gm_c12_csr_t0010.ai 50 X O H K I J a.Show that if you form a new figure by connecting the tangents to the circle at H, I, J, and K, the new figure is also a square. 13. In parallelogram ABCD below, DB is 15. What is DE? b.The inscribed square and the square formed by the tangents are similar. What is the scale geom12_gm_c12_csr_t0013.ai factor of the similar figures? Z B A E geom12_gm_c12_csr_t0011.ai C D c.Let a regular polygon with n sides be inscribed in a circle. Do the tangent lines at the vertices of the polygon form another regular polygon with n sides? Explain. 14. The measure of the vertex angle of an isosceles triangle is 112. What is the measure of a base angle? hsm11gmse_07cu_t05664 PearsonTEXAS.com 515