Apply Other Angle Relationships in Circles

Transcription

1 0.5 pply Other ngle elationships in ircles efore You found the measures of angles formed on a circle. Now You will find the measures of angles inside or outside a circle. Why So you can determine the part of arth seen from a hot air balloon, as in x. 25. ey Vocabulary chord, p. 65 secant, p. 65 tangent, p. 65 You know that the measure of an inscribed angle is half the measure of its intercepted arc. This is true even if one side of the angle is tangent to the circle. THOM or Your Notebook THOM 0. If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc. 2 roof: x. 27, p. 685 m 5 } 2 m m 2 5 }2 m X M L ind angle and arc measures Line m is tangent to the circle. ind the measure of the red angle or arc. a. b. m 308 m 258 L Solution a. m 5 } 2 (308) b. ml 5 2(258) GUI TI for xample ind the indicated measure.. m 2. m ST 3. m XY 208 T 988 S Y 808 X 680 hapter 0 roperties of ircles

2 INTSTING LINS N ILS If two lines intersect a circle, there are three places where the lines can intersect. on the circle inside the circle outside the circle You can use Theorems 0.2 and 0.3 to find measures when the lines intersect inside or outside the circle. THOMS THOM 0.2 ngles Inside the ircle Theorem If two chords intersect inside a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle. roof: x. 28, p. 685 or Your Notebook 2 m 5 } m 2 m 2, m 2 5 } m 2 m 2 THOM 0.3 ngles Outside the ircle Theorem If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. 2 m 5 } 2 m 2 m 2 m 2 5 } 2 m Q 2 m 2 m 3 5 }2 m XY 2 m WZ 2 roof: x. 29, p. 685 W 3 Z X Y X M L 2 ind an angle measure inside a circle ind the value of x. 308 M Solution L The chords } L and } M intersect inside the circle } 2 mm ml 2 Use Theorem } 2 ( ) x 5 43 Substitute. Simplify. 0.5 pply Other ngle elationships in ircles 68

3 X M L 3 ind an angle measure outside a circle ind the value of x. Solution The tangent ] and the secant ] intersect outside the circle. m 5 } m 2 m 2 Use Theorem } 2 ( ) x 5 5 Substitute. Simplify. X M L 4 Solve a real-world problem SIN The Northern Lights are bright flashes of colored light between 50 and 200 miles above arth. Suppose a flash occurs 50 miles above arth. What is the measure of arc, the portion of arth from which the flash is visible? (arth s radius is approximately 4000 miles.) 450 mi 4000 mi Solution ecause } and } are tangents, } } and } }. lso, } > } and } > }. So, n > n by the Hypotenuse-Leg ongruence Theorem, and >. Solve right n to find that m ø So, m ø 2(74.58) ø 498. Let m 5. m 5 } 2 m 2 m 2 Use Theorem 0.3. Not drawn to scale VOI OS ecause the value for m is an approximation, use the symbol ø instead of ø } 2 [( ) 2 ] Substitute. x ø 3 Solve for x. c The measure of the arc from which the flash is visible is about 38. at classzone.com GUI TI for xamples 2, 3, and 4 ind the value of the variable. 4. y H 308 G a T 3 S U 682 hapter 0 roperties of ircles

4 0.5 XISS SILL TI HOMWO Y 5 WO-OUT SOLUTIONS on p. WS for xs. 3, 9, and 23 5 STNIZ TST TI xs. 2, 6, 3, 5, 9, and 26. VOULY opy and complete: The points,,, and are on a circle and ] intersects ] at. If m 5 } 2 m 2 m 2, then is? (inside, on, or outside) the circle. 2. WITING What does it mean in Theorem 0.2 if m 5 08? Is this consistent with what you learned in Lesson 0.4? xplain your answer. XML on p. 680 for xs. 3 6 INING MSUS Line t is tangent to the circle. ind the indicated measure. 3. m 4. m 5. m t t 658 t MULTIL HOI The diagram at the right is not drawn to scale. } is any chord that is not a diameter of the circle. Line m is tangent to the circle at point. Which statement must be true? x 90 x 90 m x 5 90 x Þ 90 INING MSUS ind the value of x. XML 2 on p. 68 for xs H G M L (2x 2 30)8 XML 3 on p. 682 for xs G S U 348 T (x 6)8 (3x 2 2)8 W V 3. MULTIL HOI In the diagram, l is tangent to the circle at. Which relationship is not true? m 5 08 m m m T S 608 l 0.5 pply Other ngle elationships in ircles 683

5 4. O NLYSIS escribe the error in the diagram below SHOT SONS In the diagram at the right, ] L is tangent to the circle and } is a diameter. What is the range of possible angle measures of L? xplain. L 6. ONNTI ILS The circles below are concentric. a. ind the value of x. b. xpress c in terms of a and b a8 b8 c8 7. INSI IL In the diagram, the circle is inscribed in nq. ind m, m G, and m G. 8. LG In the diagram, ] is tangent to (. ind m. 408 G WITING oints and are on a circle and t is a tangent line containing and another point. a. raw two different diagrams that illustrate this situation. b. Write an equation for m in terms of m for each diagram. c. When will these equations give the same value for m? HLLNG ind the indicated measure(s). 20. ind m if mwzy ind m and m. Z W Y X G 58 H WO-OUT SOLUTIONS on p. WS 5 STNIZ TST TI

6 OLM SOLVING VIO OING In the diagram at the right, television cameras are positioned at,, and to record what happens on stage. The stage is an arc of (. Use the diagram for xercises ind m, m, and m. 23. The wall is tangent to the circle. ind x without using the measure of. 24. You would like amera to have a 308 view of the stage. Should you move the camera closer or further away from the stage? xplain. XML 4 on p. 682 for x HOT I LLOON You are flying in a hot air balloon about.2 miles above the ground. Use the method from xample 4 to find the measure of the arc that represents the part of arth that you can see. The radius of arth is about 4000 miles. 26. XTN SONS cart is resting on its handle. The angle between the handle and the ground is 48 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? xplain. 27. OVING THOM 0. The proof of Theorem 0. can be split into three cases. The diagram at the right shows the case where } contains the center of the circle. Use Theorem 0. to write a paragraph proof for this case. What are the other two cases? (Hint: See xercises 3 33 on page 678.) raw a diagram and write plans for proof for the other cases. 28. OVING THOM 0.2 Write a proof of Theorem GIVN c hords } and } intersect. OV c m 5 } 2 m m OVING THOM 0.3 Use the diagram at the right to prove Theorem 0.3 for the case of a tangent and a secant. raw }. xplain how to use the xterior ngle Theorem in the proof of this case. Then copy the diagrams for the other two cases from page 68, draw appropriate auxiliary segments, and write plans for proof for these cases. 0.5 pply Other ngle elationships in ircles 685

7 30. OO Q and are points on a circle. is a point outside the circle. } Q and } are tangents to the circle. rove that } Q is not a diameter. 3. HLLNG 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 3 centimeters and the pulleys each have a radius of 5 centimeters. What percent of the circumference of the bottom pulley is not touching the rope? MIX VIW lassify the dilation and find its scale factor. (p. 626) VIW repare for Lesson 0.6 in xs Use the quadratic formula to solve the equation. ound decimal answers to the nearest hundredth. (pp. 64, 883) 34. x 2 7x x 2 2 x x x 37. x 2 6x x 9 5 2x x 2 3x QUIZ for Lessons ind the value(s) of the variable(s).. m 5 z8 (p. 672) 2. m GH 5 z8 (p. 672) 3. m L 5 z8 (p. 672) G y (x y)8 H M L (p. 680) (p. 680) (p. 680) 7. MOUNTIN You are on top of a mountain about.37 miles above sea level. ind the measure of the arc that represents the part of arth that you can see. arth s radius is approximately 4000 miles. (p. 680) 686 XT TI for Lesson 0.5, p. 95 ONLIN QUIZ at classzone.com

8 MIX VIW of roblem Solving Lessons MULTI-ST OLM n official stands 2 meters from the edge of a discus circle and 3 meters from a point of tangency. STT TST TI classzone.com 4. XTN SONS The Navy ier erris Wheel in hicago is 50 feet tall and has 40 spokes. 2 m 3 m a. ind the radius of the discus circle. b. How far is the official from the center of the discus circle? 2. GI NSW In the diagram, } XY > } YZ and m XQZ ind m YZ in degrees. Y X Z 3. MULTI-ST OLM wind turbine has three equally spaced blades that are each 3 feet long. a. ind the measure of the angle between any two spokes. b. Two spokes form a central angle of 728. How many spokes are between the two spokes? c. The bottom of the wheel is 0 feet from the ground. ind the diameter and radius of the wheel. xplain your reasoning. 5. ON-N raw a quadrilateral inscribed in a circle. Measure two consecutive angles. Then find the measures of the other two angles algebraically. 6. MULTI-ST OLM Use the diagram. y 938 L M 358 N a. What is the measure of the arc between any two blades? b. The highest point reached by a blade is 36 feet above the ground. ind the distance x between the lowest point reached by the blades and the ground. x c. What is the distance y from the tip of one blade to the tip of another blade? ound your answer to the nearest tenth. a. ind the value of x. b. ind the measures of the other three angles formed by the intersecting chords. 7. SHOT SONS Use the diagram to show that m 5 y8 2. y8 Mixed eview of roblem Solving 687