4.7 Triangle Inequalities

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1 age 1 of riangle Inequalities Goal Use triangle measurements to decide which side is longest and which angle is largest. he diagrams below show a relationship between the longest and shortest sides of a triangle and the largest and smallest angles. largest angle shortest side longest side smallest angle OS 4.10 and 4.11 heorem 4.10 Words If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. Symbols If >, then ma > ma. heorem 4.11 Words If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. Symbols If ma > ma, then >. 40 XL 1 Order ngle easures ame the angles from largest to smallest V > U, so mau > mav. lso, U > UV, so mav > ma. SW he order of the angles from largest to smallest is au, av, a. U 4 V 212 hapter 4 riangle elationships

2 age 2 of 7 IStudent elp I L S S Z O. O O XLS ore examples at classzone.com XL 2 Order Side Lengths ame the sides from longest to shortest ma > ma, so >. lso, ma > ma, so >. SW he order of the sides from longest to shortest is &*, &*, &*. Order ngle easures and Side Lengths ame the angles from largest to smallest. 1. L S 42 U 47 ame the sides from longest to shortest G Segments of a riangle ot every group of three segments can be used to form a triangle. he lengths of the segments must have the following relationship. O 4.12 riangle Inequality Words he sum of the lengths of any two sides of a triangle is greater than the length of the third side. Symbols > > > 4.7 riangle Inequalities 21

3 age of 7 XL Use the riangle Inequality an the side lengths form a triangle? xplain. a.,, 9 b.,, 8 c.,, 7 a. b. c hese lengths do not form a triangle, because < 9. hese lengths do not form a triangle, because 8. hese lengths do form a triangle, because > 7, 7 >, and 7 >. Use the riangle Inequality an the side lengths form a triangle? xplain. 7., 7, 1 8., 9, , 1, xercises Guided ractice Vocabulary heck Skill heck 1. omplete the statement: he symbol > means?, and the symbol < means?. 2. ame the smallest angle. ame the longest side of. of In xercises 4 and, use the figure shown at the right. omework elp xample 1: xs , 18 24, 7, 8 xample 2: xs. 1 17, 24 1, 7, 8 xample : xs. 2, 2, ame the smallest and largest angles of.. ame the shortest and longest sides of. an the side lengths form a triangle? xplain , 2, 7., 10, , 1, , 8, , 9, 1 11.,, hapter 4 riangle elationships

4 age 4 of 7 ractice and pplications xtra ractice See p. 82. omparing ngle easures ame the smallest and largest angles of the triangle G omparing Side Lengths ame the shortest and longest sides of the triangle S Ordering ngles ame the angles from largest to smallest. 18. L S X 7 Y 2. itchen esign W esign In xercises 24 and 2, use the following information. he term kitchen triangle refers to the imaginary triangle formed by the refrigerator, the sink, and the stove. he distances shown are measured in feet. I IGLS or ease of movement among appliances, the perimeter of an ideal kitchen triangle should be less than 22 feet and more than 1 feet. 24. What is wrong with the labels on the kitchen triangle? 2. an a kitchen triangle have the following side lengths: 9 feet, feet, and feet? xplain why or why not. SI SOV 4 ft 82.4 ft 8. ft IGO 4.7 riangle Inequalities 21

5 age of 7 Ordering Sides ame the sides from longest to shortest G G 8 2 rror nalysis xplain why the side lengths given with the triangles are not correct XL Is it possible to draw a triangle that has side lengths of 4,, and? If so, draw the triangle. Use the riangle Inequality Yes, these side lengths satisfy the riangle Inequality: 4 >, > 4, and 4 >. So, it is possible to draw the triangle, as shown below. 1 ark &* of length 4 cm on a line. hen draw an arc of radius cm with center at. 2 raw an arc of radius cm with center at. ark the intersection of the two arcs as. has side lengths of 4 cm, cm, and cm hapter 4 riangle elationships Using the riangle Inequality etermine whether it is possible to draw a triangle with the given side lengths. If so, draw the triangle. 4. 4, 7, , 12, , 9, 0

6 age of 7 Student elp VISUL SGY In xs. 7 and 8, draw a sketch with measurements that are roughly correct, as shown on p Visualize It! Sketch a triangle and label it with the given angle measures and side lengths. 7. ngles: 9, 4, 7 8. ngles: 1, 1, 0 Sides: 1 cm, 9.7 cm, 11. cm Sides: 7.1 cm, 2. cm, cm 9. aking a Shortcut Suppose you are walking south on the sidewalk of ine Street. When you reach leasant Street, you cut across the empty lot to go to the corner of Oak ill venue and Union Street. xplain why this route is shorter than staying on the sidewalks. Oak ill ve. leasant St. Union St. ine St. You be the udge 40. Suppose you are camping. You decide to hike 4. miles northwest and then turn and hike 1.8 miles east. Your friend tells you that you are about one and a half miles from camp. Is your friend right? xplain why or why not. Logical easoning In xercises 41 4, use the figure shown and the given information. y adjusting the length of the boom lines from to, the operator of the crane shown can raise and lower the boom. 0 ft Suppose the mast &* is 0 feet long and the boom &* is 100 feet long. 100 ft 41. Is the boom raised or lowered when the boom lines are shortened? 42. must be less than? feet. 4. s the boom is raised or lowered, is a ever larger than a? xplain. Standardized est ractice 44. ulti-step roblem You are given an 18-inch piece of wire. You want to bend the wire to form a triangle so that the length of each side is a whole number. a. Sketch four possible isosceles triangles and label each side length. b. Sketch a possible acute scalene triangle. c. Sketch a possible obtuse scalene triangle. d. List three combinations of segment lengths with a sum of 18 that will not produce triangles. 4.7 riangle Inequalities 217

7 age 7 of 7 ixed eview Identifying arts of a riangle In xercises 4 48, use the figure shown to complete the statement. (Lessons 4.1, 4., 4.4) 4.? is the hypotenuse of S. 4. In S, &* is the side? as. S 47. he legs of S are? and?. 48.? is the base of S. inding easures ind the measure of the numbered angle. (Lesson 4.2) lgebra Skills Solving roportions Solve the proportion. (Skills eview, p. 0) x x 4. x x. 8 x x Quiz Use the side lengths to classify the triangle as acute, right, or obtuse. (Lesson 4.) 1., 11, , 7, 1. 18, 80, 82 is the centroid of L. ind and. (Lesson 4.) 4. =. = 9. = 0 L L L ame the sides from longest to shortest. (Lesson 4.7) 7. L hapter 4 riangle elationships

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