14.3 Tangents and Circumscribed Angles

Transcription

1 Name lass Date 14.3 Tangents and ircumscribed ngles Essential uestion: What are the key theorems about tangents to a circle? Explore G.5. Investigate patterns to make conjectures about geometric relationships, including... special segments and angles of circles. lso G.12. Investigating the Tangent-adius Theorem esource Locker tangent is a line in the same plane as a circle that intersects the circle in exactly one point. The point where a tangent and a circle intersect is the point of tangency. In the figure, the line is tangent to circle, and point is the point of tangency. You can use a compass and straightedge to construct a circle and a line tangent to it. Use a compass to draw a circle. Label the center. Mark a point on the circle. Using a straightedge, draw a tangent to the circle through point. Mark a point at a different position on the tangent line. Use a straightedge to draw the radius _. Houghton Mifflin Harcourt ublishing ompany Use a protractor to measure. ecord the result in the table. epeat the process two more times. Make sure to vary the size of the circle and the location of the point of tangency. Vehicle 1 ircle 1 ircle 2 ircle 3 Measure of Module Lesson 3

2 eflect 1. Make a onjecture Examine the values in the table. Make a conjecture about the relationship between a tangent line and the radius to the point of tangency. 2. Discussion Describe any possible inaccuracies related to the tools you used in this Explore. Explain 1 roving the Tangent-adius Theorem The Explore illustrates the Tangent-adius Theorem. Tangent-adius Theorem If a line is tangent to a circle, then it is perpendicular to a radius drawn to the point of tangency. roof of Tangent-adius Theorem Given: Line m is tangent to circle at point. rove: m m You will complete a proof of this theorem in Example 1 on the next page. Houghton Mifflin Harcourt ublishing ompany Module Lesson 3

3 Example 1 omplete the proof of the Tangent adius Theorem. m Use an indirect proof. ssume that is not perpendicular to line m. There must be a point on line m such that m. If m, then is a triangle, and > because is the of the right triangle. Since line m is a tangent line, it can intersect circle at only point other points, and all of line m are in the This means point is in the is a of circle. of the circle. of the circle. You can conclude that < because This contradicts the initial assumption that a point exists such that _ m, because that meant that >. Therefore, the assumption is and must be perpendicular to line m. eflect 3. oth lines in the figure are tangent to the circle, and _ is a diameter. What can you conclude about the tangent lines? The converse of the Tangent-adius Theorem is also true. You will be asked to prove this theorem as an exercise. Houghton Mifflin Harcourt ublishing ompany onverse of the Tangent-adius Theorem If a line is perpendicular to a radius of a circle at a point on the circle, then it is tangent to the circle at that point on the circle. Module Lesson 3

4 Explain 2 onstructing Tangents to a ircle From a point outside a circle, two tangent lines can be drawn to the circle. Example 2 Use the steps to construct two tangent lines from a point outside a circle. Use a compass to draw a circle. Label the center. Mark a point X outside the circle and use a straightedge to draw _ X. Use a compass and straightedge to construct the midpoint of _ X and label the midpoint M. Use a compass to construct a circle with center M and radius M. Label the points of intersection of circle and circle M as and. Use a straightedge to draw X and X. oth lines are tangent to circle. eflect 4. How can you justify that X (or X ) is a tangent line? (Hint: Draw _ on the diagram.) 5. Draw _ and _ on the diagram. onsider quadrilateral X. State any conclusions you can reach about the measures of the angles of X. Houghton Mifflin Harcourt ublishing ompany Module Lesson 3

5 Explain 3 roving the ircumscribed ngle Theorem circumscribed angle is an angle formed by two rays from a common endpoint that are tangent to a circle. ircumscribed ngle Theorem circumscribed angle of a circle and its associated central angle are supplementary. Example 3 rove the ircumscribed ngle Theorem. Given: X is a circumscribed angle of circle. rove: X and are supplementary. X Since X is a circumscribed angle of circle, _ X and _ X are X are by the. to the circle. Therefore, X and In quadrilateral X, the sum of the measures of its four angles is. Since m X + m X =, this means m X + m = =. So, X and are supplementary by the. Houghton Mifflin Harcourt ublishing ompany eflect 6. Is it possible for quadrilateral X to be a parallelogram? If so, what type of parallelogram must it be? If not, why not? Module Lesson 3

6 Elaborate 7. KM and KN are tangent to circle. Explain how to show that KM KN, using congruent triangles. M K N 8. Essential uestion heck-in What are the key theorems regarding tangent lines to a circle? Houghton Mifflin Harcourt ublishing ompany Module Lesson 3

7 Evaluate: Homework and ractice Use the figure for Exercises 1 2. You use geometry software to construct a tangent to circle O at point X on the circle, as shown in the diagram. X Online Homework Hints and Help Extra ractice O Y 1. What do you expect to be the measure of OXY? Explain. 2. Suppose you drag point X so that is in a different position on the circle. Does the measure of OXY change? Explain. 3. Make a onjecture You use geometry software to construct circle, diameters _ and _ D, and lines m and n which are tangent to circle at points D and, respectively. Make a conjecture about the relationship of the two tangents. Explain your conjecture. D n Houghton Mifflin Harcourt ublishing ompany Image redits: NS 4. In the figure, _ is tangent to circle at point. What is m? Explain your reasoning. 5. epresent eal-world roblems The International Space Station orbits Earth at an altitude of about 240 miles. In the diagram, the Space Station is at point E. The radius of Earth is approximately 3960 miles. To the nearest ten miles, what is EH, the distance from the space station to the horizon? 59 D E? H 3960 mi Module Lesson 3

8 Multi-Step Find the length of each radius. Identify the point of tangency, and write the equation of the tangent line at that point y y x x In the figure, S = 5, T = 12, and T is tangent to radius _ with the point of tangency at. Find T S T The segments in each figure are tangent to the circle at the points shown. Find each length. 9. 2x x D S T y y 2 7 Houghton Mifflin Harcourt ublishing ompany Module Lesson 3

9 11. Justify easoning Suppose you construct a figure with tangent to circle at and S tangent to circle at S. Make a conjecture about and. Justify your reasoning. S 12. is tangent to circle at and S is tangent to circle at S. Find m. 42 S 13. is tangent to circle at and S is tangent to circle at S. Find m. x S 3x Houghton Mifflin Harcourt ublishing ompany Module Lesson 3

10 is tangent to circle O at and is tangent to circle O at, and m = 56. Use the figure to find each measure m O F O G 15. m OGF 16. Which statements correctly relate D and? Select all that apply.. D and are complementary.. D and are supplementary.. D and are congruent. D. D and are right angles. E. The sum of the measures of D and is 180. F. It is impossible to determine a relationship between D and. D 17. ritical Thinking Given a circle with diameter _, is it possible to construct tangents to and from an external point X? If so, make a construction. If not, explain why it is not possible. KJ is tangent to circle at J, KL is tangent to circle at L, and m ML = Find m M. 19. Find m MJ. M 138 L J 40 K Houghton Mifflin Harcourt ublishing ompany Module Lesson 3

11 H.O.T. Focus on Higher Order Thinking 20. Justify easoning rove the converse of the Tangent-adius Theorem. Given: Line m is in the plane of circle, is a point of circle, and _ m rove: m is tangent to circle at. m 21. Draw onclusions grapic designer created a preliminary sketch for a company logo. In the figure, and D are tangent to circle and >. What type of quadrilateral is figure D that she created? Explain. D 22. Explain the Error In the given figure, and are tangents. student was asked to find m S. ritique the student s work and correct any errors. Since is a circumscribed angle, and are supplementary. So m = 110. Since S, m S = S x T Houghton Mifflin Harcourt ublishing ompany 23. Given circle O and points and, construct a triangle that is circumscribed around the circle. O Module Lesson 3

12 Lesson erformance Task communications satellite is in a synchronous orbit 22,000 miles above Earth s surface. oints and D in the figure are points of tangency of the satellite signal with the Earth. They represent the greatest distance from the satellite at which the signal can be received directly. oint is the center of the Earth. 1. Find distance. ound to the nearest mile. Explain your reasoning. 2. m = 9. If the circumference of the circle represents the Earth s equator, what percent of the Earth s equator is within range of the satellite s signal? Explain your reasoning. 22,000 mi 3. How much longer does it take a satellite signal to reach point than it takes to reach point E? Use 186,000 mi/sec as the speed of a satellite signal. ound your answer to the nearest hundredth. E 4,000 mi D 4. The satellite is in orbit above the Earth s equator. long with the point directly below it on the Earth s surface, the satellite makes one complete revolution every 24 hours. How fast must it travel to complete a revolution in that time? You can use the formula = 2πr to find the circumference of the orbit. ound your answer to the nearest whole number. Houghton Mifflin Harcourt ublishing ompany Module Lesson 3