Ready To Go On? Skills Intervention 11-1 Lines That Intersect Circles

Transcription

1 Name ate lass STION 11 Ready To Go On? Skills Intervention 11-1 Lines That Intersect ircles ind these vocabulary words in Lesson 11-1 and the Multilingual Glossary. Vocabulary interior of a circle exterior of a circle chord secant tangent of a circle point of tangency congruent circles concentric circles tangent circles common tangent Identifying Lines and Segments That Intersect ircles Identify each line or segment that intersects. t chords: tangent: t t radii: secant: diameter: t t t Tangents of ircles ind MN. What is OP? What is ON? O 9 cm M O M M N O N M N M N Use the Pythagorean Theorem. Substitute known values. Simplify. P 6 cm N M N 144 Subtract 81 from both sides. MN Take the square root of both sides. opyright by Holt, Rinehart and Winston. 158 Holt Geometry

2 Name ate lass STION 11 Ready To Go On? Problem Solving Intervention 11-1 Lines that Intersect ircles You can use lines and circles to solve problems about the height of buildings, the horizon, and the arth. The tallest building in the United States is the Sears Tower in hicago. What is the distance from the top of this 1450-ft building to the arth s horizon to the nearest mile? (Hint: 580 ft 1 mi; radius of the arth 4000 mi) Understand the Problem 1. What are you to determine?. In what units will your answer be measured? Make a Plan Use the sketch at the right to help you solve this problem. Note that the sketch is not drawn to scale. 3. How is _ related to _? 4. What kind of angle is? 5. What does represent? 6. What length do you need to find? 4000 mi Solve 7. onvert the height of the Sears Tower to miles: mi ind. 9. pply the Pythagorean Theorem: Substitute. Solve. Round to the nearest mile. mi 10. What is the distance from the top of the Sears Tower to the arth s horizon? Look ack 11. heck by substituting your answer from xercise 10 into the Pythagorean Theorem. Is ? opyright by Holt, Rinehart and Winston. 159 Holt Geometry

3 Name ate lass STION 11 Ready To Go On? Skills Intervention 11- rcs and hords ind these vocabulary words in Lesson 11- and the Multilingual Glossary. Vocabulary central angle arc minor arc major arc semicircle adjacent arcs congruent arcs Using the rc ddition Postulate ind each measure. Q P. m QP m QT m PT mqot 90. mtop 4. R O 4 T m QP m PT m QT Use the rc ddition Postulate. m QP Substitute known values. S m QP Subtract 4 from both sides.. m RTP m RST m TP m RTP m RST Use the rc ddition Postulate. RST is a. m TP mtop 4. m RTP Substitute known values. m RTP Simplify. P Using Radii and hords ind PN. Step 1 raw radius _ MN. MN Radii of an arc are congruent. M 7 Q 3 O N Step Use the Pythagorean Theorem. MQ QN MN QN Substitute known values. QN Simplify. QN Subtract 49 from both sides. QN Take the square root of both sides. Step 3 ind PN. PN QN _ MQ _ PN so _ MQ bisects _ PN. opyright by Holt, Rinehart and Winston. 160 Holt Geometry

4 Name ate lass STION 11 Ready To Go On? Skills Intervention 11-3 Sector rea and rc Length ind these vocabulary words in Lesson 11-3 and the Multilingual Glossary. Vocabulary sector of a circle segment of a circle arc length inding the rea of a Sector ind the area of sector. Give your answer in terms of and rounded to the nearest hundredth. r m 360 Use the formula for area of a sector m ( ) 360 Substitute for r and for m. Simplify. m inding rc Length ind each arc length. Give your answer in terms of and rounded to the nearest hundredth. R 5 in.. RS L r m 360 Use the formula for arc length. 135 T 360 Substitute for r and for m. S Simplify in.. a semicircle in a circle with diameter 1 ft r r d L r m 360 Use the formula for arc length. 360 Substitute for r and for m. Simplify. in. opyright by Holt, Rinehart and Winston. 161 Holt Geometry

5 Name ate lass STION 11 Ready To Go On? Problem Solving Intervention 11-3 Sector rea and rc Length You can find the area of a sector to solve problems about designing security systems. The beam from a security light is visible for 37 m. To the nearest square meter, what is the area covered by the beam as it sweeps in an arc of 160? Understand the Problem 1. What does the 37-m beam represent?. What information do you need to answer the question? m Make a Plan 3. What formula do you need to solve this problem? Solve 4. Substitute the given values into the sector area formula Simplify and round your answer to the nearest square meter. m 6. What is the area covered by the beam as it sweeps in an arc of 160? Look ack 7. Since 160 is slightly less than 180, check to see if your answer is reasonable by finding the area of a semicircle with radius 37 m. Round your answer to the nearest square meter. 8. Is your answer from xercise 6 slightly less than your answer from xercise 7? Is your answer reasonable? opyright by Holt, Rinehart and Winston. 16 Holt Geometry

6 Name ate lass STION 11 Ready To Go On? Quiz 11-1 Lines that Intersect ircles Identify each line or segment that intersects each circle. 1.. M N L Q P 3. The mpire State uilding in New York ity is 150 ft tall. What is the distance from the top of this building to the horizon to the nearest mile? (Hint: 580 ft 1 mi; radius of arth 4000 mi) 11- rcs and hords ind each measure. 4. m TU 5. m TYV X W H Y T 34 U V I G 6. m H 7. m HG opyright by Holt, Rinehart and Winston. 163 Holt Geometry

7 Name ate lass STION 11 Ready To Go On? Quiz continued ind each length to the nearest tenth LN L OP = O M P N 11-3 Sector rea and rc Length 10. security camera can monitor a circular area with radius 36 ft. To the nearest square foot, what is the area that can be monitored as the camera rotates through an angle of 110? ft ind each arc length. Give your answer in terms of and rounded to the nearest tenth. 11. ST 1. XY 10 S X 40 T 7 m 5.3 ft Y 13. an arc with measure 58 in a circle with diameter 18 cm 14. a semicircle with diameter 64 in. opyright by Holt, Rinehart and Winston. 164 Holt Geometry

8 Name ate lass STION 11 Ready To Go On? nrichment Lines and rcs in ircles Use your knowledge of lines and arcs in circles to answer each question. 1. The picture at the right shows the part of a windshield covered by the windshield wiper. The arc of the wiper is 115. The length of the entire wiper is 18 in. and the blade is 1 in. ind the area of the windshield covered by the wiper. Round your answer to the nearest inch ind the area of the shaded region in the figure at the right. Round your answer to the nearest hundredth. 9 cm Janice is planting a garden in the shape of the sector at the right. How much edging should she buy? Round your answer to the nearest foot ft 4. The straight edges of the sector at the right are joined together to form a cone. What is the circumference of the base of the cone? 4 in. 140 What is the radius base of the cone? 5. clock has a minute hand that is 7 in. long. How far has the tip of the minute hand traveled between 10:5.M. and 10:50.M.? Round your answer to the nearest tenth of an inch. opyright by Holt, Rinehart and Winston. 165 Holt Geometry

9 Name ate lass STION 11 ind these vocabulary words in Lesson 11-4 and the Multilingual Glossary. Vocabulary Ready To Go On? Skills Intervention 11-4 Inscribed ngles inscribed angle intercepted arc subtend K inding Measures of rcs and Inscribed ngles ind each measure.. mkjl 10 L 4 J. LM mkjl 1 m Use the Inscribed ngle Theorem. 1 Substitute for m KL and simplify. mljm 1 LM Use the Inscribed ngle Theorem. K M 10 L K 4 M J 4 1 LM Substitute for mljm. LM Multiply both sides by. 10 L M 4 J inding ngle Measures in Inscribed Triangles ind each measure.. m is a angle. is inscribed in a. m. efinition of a angle. 4. m m 1 Use the Inscribed ngle Theorem. 4 1 Substitute for m. 4 m Multiply both sides by. m is a. m m m Use the rc ddition Postulate. m Substitute for m and 180 for m and simplify. 4 opyright by Holt, Rinehart and Winston. 166 Holt Geometry

10 Name ate lass STION 11 Ready To Go On? Skills Intervention 11-5 ngle Relationships in ircles Using Tangent-Secant and Tangent-hord ngles ind myxz. The measure of the angle formed by a tangent and a secant is half the measure of its intercepted arc. Y myxz 1 m X 186 myxz 1 Substitute for m and simplify. Z inding ngle Measures Inside a ircle ind mg. The measure of the angle formed by two secants or chords is half the measure of the sum of the intercepted arcs. mg 1 m m 1 Substitute for m G and for m. 134 G 16 1 Simplify. inding Measures Using Tangents and Secants ind m. m 360 The sum of non-overlapping arcs of a circle is 360. m 1 m m The measure of the angle formed by two tangents is half the measure of the difference of its intercepted arcs Substitute for m and for m. 1 ( ) Simplify. opyright by Holt, Rinehart and Winston. 167 Holt Geometry

11 Name ate lass STION 11 Ready To Go On? Problem Solving Intervention 11-5 ngle Relationships in ircles ircles and angle relationships are used in a variety of design applications. carpenter is building a decorative brick wall in front of a circular fountain. He plans to build the wall so that m ZX 78. What should be the measure of ZYX? Understand the Problem 1. How are YZ and YX related to the circle? Y Z X W. What are you suppose to determine? Make a Plan 3. How can you find the measure of ZWX? 4. How is mzyx related to the arcs of the circle? Solve 5. ind m ZWX : m ZWX ind mzyx: mzyx 1 m m The measure of ZYX should be. Look ack 8. The angle formed by two tangents to a circle and the measure of the minor arc between the two points of tangency are supplementary. heck your answer in xercise 7 by determining if mzyx and m ZX are supplementary. Show your work. opyright by Holt, Rinehart and Winston. 168 Holt Geometry

12 Name ate lass STION 11 Ready To Go On? Skills Intervention 11-6 Segment Relationships in ircles ind these vocabulary words in Lesson 11-6 and the Multilingual Glossary. Vocabulary secant segment external secant segment tangent segment pplying the hord-hord Product Theorem ind the value of a and the length of each chord segment. VZ ZY Use the hord-hord Product Theorem. V 6( ) a Substitute for ZX and for ZY. 6 W a Z 1 8 X 1a Multiply. 4 a ivide both sides by. Y WZ ind the length of each chord segment: WY Use the Segment ddition Postulate. VX Use the Segment ddition Postulate G 7 x pplying the Secant-Secant Product Theorem ind the value of x and the length of each secant segment. G Use the Secant-Secant Product Theorem. (18 7) (x 15) Substitute known values. 18 (x 15)15 Simplify. 15x ind the secant lengths: pply the istributive Property. 15x Subtract from both sides. x ivide by on both sides. Use the Segment ddition Postulate. Use the Segment ddition Postulate. opyright by Holt, Rinehart and Winston. 169 Holt Geometry

13 Name ate lass STION 11 Ready To Go On? Problem Solving Intervention 11-6 Segment Relationships in ircles You can use relationships between the segments in circles to find missing measurements. The first erris wheel was built in 189. If 19 ft, 80 ft and _ is the perpendicular bisector of _, find the diameter of the original erris wheel. Round your answer to the nearest foot. Understand the Problem 1. To find the diameter of the erris wheel, you must first find the.. Name three radii in the picture. 3. To solve this problem, what theorem must you use? Make a Plan 4. Name a right triangle in the figure. 5. Name the legs of the right triangle. 6. Name the hypotenuse of the right triangle. Solve 7. Since _ is the perpendicular bisector of _, what is true about _ and _? 8. ind. 1 1 ft 9. Substitute the known values into the Pythagorean Theorem to find. 10. ind the diameter of the original erris wheel. d r ( ) ft 11. The diameter of the original erris wheel was ft. ft Look ack 1. Substitute your solution from xercise 9 into the Pythagorean Theorem. Is ? opyright by Holt, Rinehart and Winston. 170 Holt Geometry

14 Name ate lass STION 11 Ready To Go On? Skills Intervention 11-7 ircles in the oordinate Plane Writing the quation of a ircle Write the equation of each circle.. M with center M(5, 3) and radius 7 (x h ) (y k ) r Write the equation for a circle. x y x y Simplify.. T that passes through (5, 3) and has center T(7, ) Use the istance ormula to find the radius: r ( x x 1 ) ( y y 1 ) Substitute for h, for k, and for r. (5 ( ) ) ( () ) Substitute known values. (1 ) ( ) Simplify. 13 Write the equation for the circle. (x h ) (y k ) r x ( ) y ( ) x y Simplify. Substitute for h, for k, and for r. inding the enter of a ircle ind the center of the circle that passes through (, 1), (3, 4), and (6, 5). 6 Step 1 Plot the given points. Step onnect,, and to form a triangle. 4 Step 3 onstruct the perpendicular bisectors of and _ and _ ind the point of intersection of the perpendicular bisectors: 4 The center of the circle that passes through,, and is. opyright by Holt, Rinehart and Winston. 171 Holt Geometry

15 Name ate lass STION 11 Ready To Go On? Problem Solving Intervention 11-7 ircles in the oordinate Plane You can use circles and the coordinate plane to find strategic locations. town is building a new school. To make sure it is centrally located, the school must be equidistant from three houses which are located on a coordinate plane at R(4, 4), S(3, 11), and T(10, 4). What are the coordinates where the school should be built? Understand the Problem 1. How should your answer be given?. What will your answer represent? Make a Plan 3. The perpendicular bisectors of a triangle intersect at a point that is Solve the Problem from each vertex. 4. Step 1 Plot the given points on the grid at right. Step onnect R, S, and T to form a triangle. Step 3 onstruct the perpendicular bisectors of _ RS and _ ST. 5. ind the point of intersection of the perpendicular bisectors. Label this point P. 4 4 y x 6. The point that is equidistant from R, S, and T and where the school should be located is. Look ack 7. Use the istance ormula to find PR, PT, and PS. 8. re R, S, and T equidistant from P? opyright by Holt, Rinehart and Winston. 17 Holt Geometry

16 Name ate lass STION Inscribed ngles ind each measure. 1. mmlp Ready To Go On? Quiz. m MN M 5 L N 88 P 3. m 4. m G G ngle Relationships in ircles ind each measure. 5. m 6. mspt R 70 Q P 00 S 94 T 7. n architect is planning security lights for the outside of a circular stadium. He wants each light to illuminate the stadium so that m JK = 7. ind mjlk. J L K opyright by Holt, Rinehart and Winston. 173 Holt Geometry

17 Name ate lass STION 11 Ready To Go On? Quiz continued 11-6 Segment Relationships in ircles ind the value of the variable and the length of each chord or secant segment x 3 4 T 5 S 4 Q P 3 x R 10. n archaeologist discovers a fragment of an ancient circular plate, shown by PQ in the figure. PQ 9. in. and RS 3 in. What was the diameter of the original plate? Round to the nearest hundredth. P S R Q 11-7 ircles in the oordinate Plane Write the equation of each circle. 11. P with center P(4, 5) and radius 6 1. Q that passes through (6, ) and that has center Q (3, ) 13. park has camping facilities located at (1, 3), (, 6), and (7, 3). The park wants to build a new ranger station that is equidistant from the three camping areas. What are the coordinates of the location where the facility should be built? opyright by Holt, Rinehart and Winston. 174 Holt Geometry

18 Name ate lass STION 11 Ready To Go On? nrichment ngles and Segments in hords Use the corresponding diagrams to answer each question. 1. In the circle at the right, m 197, m 163, and m 10. ind the following: m m m m m. photographer is standing at point taking a group picture at a high school reunion. His camera has a 73 field of vision. The radius of the circle is 14 ft. What is the length of _, the line along which the group of people are standing? Round your answer to the nearest tenth of a foot Write a paragraph proof to show that the sum of an angle formed by two tangents to a circle and the minor arc that is intercepted, is ind x and the length of the secant segment. 4 x 6 opyright by Holt, Rinehart and Winston. 175 Holt Geometry