Topic 12 Theorems About Circles TOPIC OVERVIEW VOCABULARY 12-1 Tangent Lines English/Spanish Vocabulary Audio Online: 12-2 Chords and Arcs English Spanish 12-3 Inscribed Angles chord, p. 492 cuerda 12-4 Angle Measures and Segment inscribed angle, p. 499 ángulo inscrito Lengths intercepted arc, p. 499 arco interceptor point of tangency, p. 486 punto de tangencia secant, p. 505 secante tangent to a circle, p. 486 tangente de un círculo DIGITAL APPS 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. HOMEWORK TUTOR APP YOUR DIGITAL Do your homework anywhere! You can access the Practice and RESOURCES Application Exercises, as well as Virtual Nerd tutorials, with this PearsonTEXAS.com 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 Earth Watch Scientists estimate that there are currently about 3000 operational man-made Math satellites orbiting the Earth. These satellites serve different purposes, from communication Act Math Act Act - - to navigation and global 3 3 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 LEARNING ANIMATIONS INTERACTIVE MATH TOOLS You’ll find definitions of math You can also access all of the These interactive math tools terms in both English and stepped-out learning animations give you opportunities to Spanish. All of the terms have that you studied in class. explore in greater depth audio support. key concepts to help build understanding. INTERACTIVE EXPLORATION STUDENT COMPANION VIRTUAL NERD You’ll have access to a robust Refer to your notes and Not sure how to do some of assortment of interactive solutions in your Student the practice exercises? Check explorations, including Companion. Remember that out the Virtual Nerd videos interactive concept explorations, your Student Companion is also for stepped-out, multi-level dynamic activitites, and topic- available as an ACTIVebook instructional support. level exploration activities. accessible on any digital device. PearsonTEXAS.com 485 12-1 Tangent Lines TEKS FOCUS VOCABULARY TEKS (12)(A) Apply theorems about circles, • Point of tangency – the point where a circle and a tangent intersect including relationships among angles, radii, • Tangent to a circle – a line in the plane of the circle that intersects chords, tangents, and secants, to solve non- the circle in exactly one point contextual problems. TEKS (1)(F) Analyze mathematical • Analyze – closely examine objects, ideas, or relationships to learn relationships to connect and communicate more about their nature mathematical ideas. 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 A tangent to a circle is a line in the plane of the circle that intersects the circle in exactly one point. B The point where a circle and a tangent intersect is the point of tangency. BA is a tangent ray, and BA3 is a tangent segment. Theorem 12-1 Theorem If . Then . < > < > If a line is tangent to a AB is tangent to }O at P AB # OP circle, then the line is perpendicular to the radius A A at the point of tangency. P P O B O B For a proof of Theorem 12-1, see the Reference section on page 683. hsm11gmse_1201_t05166 hsm11gmse_1201_t05174 486 Lesson 12-1 Tangent Lines Theorem 12-2 Theorem If . Then . < > < > If a line in the plane of a AB # OP at P AB is tangent to }O circle is perpendicular to A a radius at its endpoint on the circle, then the line is P tangent to the circle. O B You will prove Theorem 12-2 in Exercise 19. Theorem 12-3 hsm11gmse_1201_t05174 Theorem If . Then . If two tangent segments to BA and BC are tangent to }O BA ≅ BC a circle share a common A endpoint outside the circle, B then the two segments are congruent. 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 117Њ xЊ M N 58 63 90 117 What kind of angle is formed by a radius Since ML and MN are tangent to }O , ∠L and ∠N are right angles (Theorem 12-1). and a tangent? LMNO is a quadrilateral. Sohsm11gmse_1201_t05167 the sum of the angle measures is 360. The angle formed is a right angle, so the m∠L + m∠M + m∠N + m∠O = 360 measure is 90. 90 + m∠M + 90 + 117 = 360 Substitute. 297 + m∠M = 360 Simplify. m∠M = 63 Solve. The correct answer is B. PearsonTEXAS.com 487 Problem 2 Finding Distance STEM Earth Science The CN Tower in Toronto, Canada, has an observation deck 447 m above ground level. About How does knowing how far is it from the observation deck to the horizon? Earth’s radius help? Earth’s radius is about 6400 km. The radius forms a right angle with a tangent line Step 1 Make a sketch. The length 447 m is about 0.45 km. from the observation T deck to the horizon. So 447 m you can use two radii, the 0.45 km tower’s height, and the E tangent to form a right triangle. C 6400 km Not to scale Step 2 Use the Pythagorean Theorem. CT2 = TE2 + CE2 (6400 0.45)2 TE2 64002 Substitute. hsm11gmse_1201_t07766+ = + (6400.45)2 = TE2 + 64002 Simplify. 40,965,760.2025 = TE2 + 40,960,000 Use a calculator. 5760.2025 = TE2 Subtract 40,960,000 from each side. 76 ≈ TE 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 What is the radius of }C? B 12 A 2 2 2 AC = AB + BC Pythagorean Theorem x 8 Why does the value x x appear on each side (x + 8)2 = 122 + x2 Substitute. C of the equation? 2 2 The length of AC, the x + 16x + 64 = 144 + x Simplify. hypotenuse, is the radius 16x = 80 Subtract x2 and 64 from each side. plus 8, which is on the left side of the equation. x = 5 Divide each side by 16. On the right side of the equation, the radius is The radius is 5. one side of the triangle. 488 Lesson 12-1 Tangent Lines Problem 4 TEKS Process Standard (1)(G) Identifying a Tangent Is ML tangent to }N at L? Explain. 25 M What information N does the diagram give you? 7 24 • LMN is a triangle. L • NM = 25, LM = 24, NL = 7 • NL is a radius. The lengths To determine ML is a tangent if ML # NL. Use the of the sides of whether ML is hsm11gmse_1201_t05175Converse of the Pythagorean Theorem to △LMN tangent to }N determine whether △LMN is a right triangle. NL2 + ML2 ≟ NM2 72 + 242 ≟ 252 Substitute. 625 = 625 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? A 10 cm D 15 cm B O F 8 cm E C How can you find the length of BC? AD = AF = 10 cm Thm 12-3: Two segments tangent to a Find the segments BD = BE = 15 cm circle from a point outside the circle are congruent to BE and EC. Then use segment CF = CE = 8 cm congruent, so they have the same length. hsm11gmse_1201_t05180 addition. 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 NLINE O H K Scan page for a Virtual Nerd™ tutorial video. O PRACTICE and APPLICATION EXERCISES M R E W O 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 1. 2. 3. 60Њ completing your homework, xЊ go to PearsonTEXAS.com. O O xЊ xЊ O 60Њ 43Њ STEM Apply Mathematics (1)(A) The circle at the right represents Earth. The radius of h d Earth is about 6400 km. Find the distance d to the horizon that a person can see r on a clearhsm11gmse_1201_t05182 day from each of the followinghsm11gmse_1201_t05183 heights h above Earth. Round your answer r to the nearest tenth of a kilometer. hsm11gmse_1201_t05184 E 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. 10 cm 9. P 15 in. x x Q hsm11gmse_1201_t05195 x 14 9 in.