10.4 Explore Inscribed Angles

Transcription

1 Investigating g eometry IIY se before esson Eplore Inscribed ngles E I compass straightedge protractor Q E I O N How are inscribed angles related to central angles? he verte of a central angle is at the center of the circle. he verte of an inscribed angle is on the circle, and its sides form chords of the circle. E X O E onstruct inscribed angles of a circle E E 2 E 3 raw a central angle se a compass to draw a circle. abel the center. se a straightedge to draw a central angle. abel it. raw points ocate three points on ( in the eterior of and label them,, and. easure angles raw,, and. hese are called inscribed angles. easure each angle. at classzone.com W O N I O N se your observations to complete these eercises. opy and complete the table. entral angle Inscribed angle Inscribed angle 2 Inscribed angle 3 Name easure???? 2. raw two more circles. epeat teps 3 using different central angles. ecord the measures in a table similar to the one above. 3. se your results to make a conjecture about how the measure of an inscribed angle is related to the measure of the corresponding central angle. 0.4 se Inscribed ngles and olygons 67

2 0.4 se Inscribed ngles and olygons efore You used central angles of circles. Now You will use inscribed angles of circles. Why? o you can take a picture from multiple angles, as in Eample 4. ey ocabulary inscribed angle intercepted arc inscribed polygon circumscribed circle n inscribed angle is an angle whose verte is on a circle and whose sides contain chords of the circle. he arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc of the angle. inscribed angle intercepted arc HEOE or Your Notebook HEOE 0.7 easure of an Inscribed ngle heorem he measure of an inscribed angle is one half the measure of its intercepted arc. roof: Es. 3 33, p. 678 m 5 } 2 m he proof of heorem 0.7 in Eercises 3 33 involves three cases. ase enter is on a side of the inscribed angle. ase 2 enter is inside the inscribed angle. ase 3 enter is outside the inscribed angle. E X E se inscribed angles ind the indicated measure in (. a. m b. m Q olution a. m 5 } 2 m 5 }2 (488) b. m Q 5 2m 5 2p ecause Q is a semicircle, m Q m Q o, m Q hapter 0 roperties of ircles

3 E X E 2 ind the measure of an intercepted arc ind m and m. What do you notice about and? 38 olution rom heorem 0.7, you know that m 5 2m 5 2(38) lso, m 5 } 2 m 5 }2 (628) o, >. INEEIN HE E Eample 2 suggests heorem 0.8. HEOE or Your Notebook HEOE 0.8 If two inscribed angles of a circle intercept the same arc, then the angles are congruent. roof: E. 34, p. 678 > E X E 3 tandardized est ractice Name two pairs of congruent angles in the figure. J > J, J > J > J, J > J J > J, J > J > J, J > J EIINE HOIE You can eliminate choices and, because they do not include the pair J > J. olution Notice that J and J intercept the same arc, and so J > J by heorem 0.8. lso, J and intercept the same arc, so they must also be congruent. Only choice contains both pairs of angles. c o, by heorem 0.8, the correct answer is. IE IE for Eamples, 2, and 3 ind the measure of the red arc or angle.. H Z Y 728 X W 0.4 se Inscribed ngles and olygons 673

4 OYON polygon is an inscribed polygon if all of its vertices lie on a circle. he circle that contains the vertices is a circumscribed circle. inscribed triangle circumscribed circles inscribed quadrilateral HEOE HEOE 0.9 If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. onversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle. roof: E. 35, p. 678 or Your Notebook m if and only if } is a diameter of the circle. E X E 4 se a circumscribed circle HOOHY Your camera has a 908 field of vision and you want to photograph the front of a statue. You move to a spot where the statue is the only thing captured in your picture, as shown. You want to change your position. Where else can you stand so that the statue is perfectly framed in this way? olution rom heorem 0.9, you know that if a right triangle is inscribed in a circle, then the hypotenuse of the triangle is a diameter of the circle. o, draw the circle that has the front of the statue as a diameter. he statue fits perfectly within your camera s 908 field of vision from any point on the semicircle in front of the statue. IE IE for Eample 4 4. WH I? In Eample 4, eplain how to find locations if you want to frame the front and left side of the statue in your picture. 674 hapter 0 roperties of ircles

5 INIE QIE Only certain quadrilaterals can be inscribed in a circle. heorem 0.0 describes these quadrilaterals. HEOE or Your Notebook HEOE 0.0 quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary., E,, and lie on ( if and only if m m 5 m E m roof: E. 30, p. 678; p. 938 E E X E 5 se heorem 0.0 ind the value of each variable. a y8 8 b. 2a8 4b8 2a8 J 2b8 olution a. Q is inscribed in a circle, so opposite angles are supplementary. m m m Q m y y b. J is inscribed in a circle, so opposite angles are supplementary. m J m m m a8 2a b8 2b a b 5 80 a 5 45 b 5 30 IE IE for Eample 5 ind the value of each variable y c (2c 2 6)8 0.4 se Inscribed ngles and olygons 675

6 0.4 EXEIE I IE HOEWO EY 5 WOE-O OION on p. W for Es., 3, and 29 5 NIZE E IE Es. 2, 6, 8, 29, and 36. OY opy and complete: If a circle is circumscribed about a polygon, then the polygon is? in the circle. 2. WIIN Eplain why the diagonals of a rectangle inscribed in a circle are diameters of the circle. EXE and 2 on pp for Es. 3 9 INIE NE ind the indicated measure. 3. m 4. m 5. m N N m 7. m 8. mwx Y W X 9. EO NYI escribe the error in the diagram of (. ind two ways to correct the error. Q 45º 00º EXE 3 ONEN NE Name two pairs of congruent angles. on p. 673 for Es W J X Y Z EXE 5 E ind the values of the variables. on p. 675 for Es y E m k J 548 4b8 3a hapter 0 roperties of ircles

7 6. IE HOIE In the diagram, is a central angle and m What is m? INIE NE In each star below, all of the inscribed angles are congruent. ind the measure of an inscribed angle for each star. hen find the sum of all the inscribed angles for each star. a. b. c. 8. IE HOIE What is the value of? E (2 40)8 (8 0)8 9. EO arallelogram Q is inscribed in (. ind m. EONIN etermine whether the quadrilateral can always be inscribed in a circle. Eplain your reasoning. 20. quare 2. ectangle 22. arallelogram 23. ite 24. hombus 25. Isosceles trapezoid 26. HENE In the diagram, is a right angle. If you draw the smallest possible circle through and tangent to }, the circle will intersect } at J and } at. ind the eact length of } J OE OIN 27. ONOY uppose three moons,, and orbit 00,000 kilometers above the surface of a planet. uppose m 5 908, and the planet is 20,000 kilometers in diameter. raw a diagram of the situation. How far is moon from moon? EXE 4 on p. 674 for E ENE carpenter s square is an -shaped tool used to draw right angles. You need to cut a circular piece of wood into two semicircles. How can you use a carpenter s square to draw a diameter on the circular piece of wood? 0.4 se Inscribed ngles and olygons 677

8 29. WIIN right triangle is inscribed in a circle and the radius of the circle is given. Eplain how to find the length of the hypotenuse. 30. OIN HEOE 0.0 opy and complete the proof that opposite angles of an inscribed quadrilateral are supplementary. IEN c ( with inscribed quadrilateral E OE c m m 5 808, m E m y the rc ddition ostulate, m E? and m m E sing the? heorem, m E 5 2m, m E 5 2m, m E 5 2m, and m 5 2m E. y the ubstitution roperty, 2m? , so?. imilarly,?. E OIN HEOE 0.7 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 eterior of the angle. In Eercises 3 33, you will prove heorem 0.7 for each of these cases. 3. ase rove ase of heorem 0.7. IEN c is inscribed in (Q. et m 5 8. oint Q lies on }. OE c m 5 } 2 m lan for roof how that n Q is isosceles. se the ase ngles heorem and the Eterior ngles heorem to show that m Q hen, show that m olve for, and show that m 5 } 2 m ase 2 se the diagram and auiliary line to write IEN and OE statements for ase 2 of heorem 0.7. hen write a plan for proof. 33. ase 3 se the diagram and auiliary line to write IEN and OE statements for ase 3 of heorem 0.7. hen write a plan for proof. 34. OIN HEOE 0.8 Write a paragraph proof of heorem 0.8. irst draw a diagram and write IEN and OE statements. 35. OIN HEOE 0.9 heorem 0.9 is written as a conditional statement and its converse. Write a plan for proof of each statement. 36. EXENE EONE In the diagram, ( and ( intersect at, and } is a diameter of (. Eplain why ] is tangent to ( WOE-O OION on p. W 5 NIZE E IE

9 HENE In Eercises 37 and 38, use the following information. You are making a circular cutting board. o begin, you glue eight inch by 2 inch boards together, as shown at the right. hen you draw and cut a circle with an 8 inch diameter from the boards. 37. } H is a diameter of the circular cutting board. Write a proportion relating J and JH. tate a theorem to justify your answer. 38. ind J, JH, and J. What is the length of the cutting board seam labeled }? J H 39. E HE o maimize thrust on a N space shuttle, engineers drill an -point star out of the solid fuel that fills each booster. hey begin by drilling a hole with radius 2 feet, and they would like each side of the star to be.5 feet. Is this possible if the fuel cannot have angles greater than 458 at its points?.5 ft 2 ft IXE EIEW EIEW repare for esson 0.5 in Es ind the approimate length of the hypotenuse. ound your answer to the nearest tenth. (p. 433) raph the reflection of the polygon in the given line. (p. 589) 43. y-ais y 5 2 y y y E ΠH ketch the image of (3, 24) after the described glide reflection. (p. 608) 46. ranslation: (, y) (, y 2 2) 47. ranslation: (, y) (, y 4) eflection: in the y-ais eflection: in y 5 4 EX IE for esson 0.4, p. 95 ONINE QIZ at classzone.com 679