Triangle Relationships

Transcription

1 hapter Triangle Relationships How is it supported? skyway 1 stories above the ground connects the Petronas Towers in Malaysia. The skyway is supported by beams that make a triangular shape. The rigid structure of a triangle is very strong. x x support beams The spires of the Petronas Towers make the building taller than hicago s Sears Tower. However, the Sears Tower has more floors: 110 compared to the 88 in the Petronas Towers. Learn More bout It You will learn more about the Petronas Towers in Exercise 26 on p

2 Who uses Triangle Relationships? WTER RESOURE MNGER Water resource managers gather information like rainfall data and water usage to study the effects of water on the environment. They use triangular structures to minimize erosion. (p. 183) ROK LIMBER The climber is using a method of rock climbing called top roping. When the ropes hanging down from the top of the rock are the same length, the angles they form at the top of the rock have the same measure. (p. 189) How will you use these ideas? Learn more about basketball plays. (p. 177) Understand how triangular structures are used to prevent erosion. (p. 183) See how rock climbers use a safety rope. (p. 189) nalyze tile patterns. (p. 190) Investigate a baseball s path during a double play. (p. 20) 171

3 hapter Study Guide PREVIEW What s the chapter about? lassifying triangles and finding their angle measures Using the Distance Formula, the Pythagorean Theorem, and its converse Showing relationships between a triangle s sides and angles Key Words equilateral, isosceles, scalene triangles, p. 173 equiangular, acute, right, obtuse triangles, p. 17 interior, exterior angles, p. 181 legs of an isosceles triangle, p. 18 base angles of an isosceles triangle, p. 18 hypotenuse, p. 192 Pythagorean Theorem, p. 192 Distance Formula, p. 19 median of a triangle, p. 207 centroid, p. 208 PREPRE hapter Readiness Quiz VISUL STRTEGY Visualize It! Take this quick quiz. If you are unsure of an answer, look at the reference pages for help. D Vocabulary heck (refer to p. 61) x 1. In the figure shown, BD &*( is the angle 30 bisector of ab. What is the value of x? B 10 B 1 20 D 30 Skill heck (refer to pp. 30, ) 2. What is the distance between P(2, 3) and Q(7, 3)? F 3 G H J 7 3. What is the midpoint of a segment with endpoints (0, 2) and B(6, )? (3, 2) B (3, 3) (, 2) D (0, 3) Drawing Triangles When you sketch a triangle, try to make the angles roughly the correct size These angles are the same in an isosceles triangle. This 80 angle is twice as big as the 0 angle. 172 hapter Triangle Relationships

4 .1 lassifying Triangles Goal lassify triangles by their sides and by their angles. Key Words equilateral, isosceles, scalene triangles equiangular, acute, right, obtuse triangles vertex triangle is a figure formed by three segments joining three noncollinear points. triangle can be classified by its sides and by its angles. 3 segments SUMMRY 3 noncollinear points LSSIFITION OF TRINGLES BY SIDES Equilateral Isosceles Scalene Triangle Triangle Triangle 3 congruent sides t least 2 No congruent sides congruent sides Stud t H lp VOBULRY TIP Equi- means equal, and -lateral means side. So, equilateral means equal sides. EXMPLE 1 lassify Triangles by Sides lassify the triangle by its sides. a. b. c Solution a. Because this triangle has 3 congruent sides, it is equilateral. b. Because this triangle has no congruent sides, it is scalene. c. Because this triangle has 2 congruent sides, it is isosceles. la ify Tria le y Side lassify the triangle by its sides lassifying Triangles 173

5 SUMMRY LSSIFITION OF TRINGLES BY NGLES Equiangular Triangle cute Triangle 3 congruent angles Right Triangle 3 acute angles Obtuse Triangle 1 right angle 1 obtuse angle Student Help I LSSZONE.OM MORE EXMPLES More examples at classzone.com EXMPLE 2 lassify Triangles by ngles and Sides lassify the triangle by its angles and by its sides. a. b. c Solution a. Because this triangle has 3 angles with measures less than 90 and 2 congruent sides, it is an acute isosceles triangle. b. Because this triangle has a right angle and no congruent sides, it is a right scalene triangle. c. Because this triangle has one angle greater than 90 and no congruent sides, it is an obtuse scalene triangle. lassify Triangles by ngles and Sides lassify the triangle by its angles and by its sides hapter Triangle Relationships

6 Student Help VOBULRY TIP The plural of vertex is vertices. vertex of a triangle is a point that joins two sides of the triangle. The side across from an angle is the opposite side. Point B is a vertex. B B&* is opposite a. EXMPLE 3 Identify the Parts of a Triangle Name the side that is opposite the angle. a. a b. ab c. a B Solution a. B &* is the side that is opposite a. b. &* is the side that is opposite ab. c. B &* is the side that is opposite a..1 Exercises Guided Practice Vocabulary heck 1. What is the difference between an obtuse triangle and an acute triangle? In Exercises 2, use the diagram. 2. Name the side opposite ap. P 3. Name the side opposite aq.. lassify the triangle by its sides. P R Skill heck lassify the triangle by its sides lassify the triangle by its angles lassifying Triangles 17

7 Practice and pplications Extra Practice See p lassifying Triangles lassify the triangle by its sides lassifying Triangles lassify the triangle by its angles Error nalysis student claims that the triangle is both obtuse and acute because it has an obtuse angle and an acute angle. What is wrong with his reasoning? This triangle is acute and obtuse. 20 lassifying Triangles lassify the triangle by its angles and by its sides. 2. B 2. E L M D F N Homework H lp Example 1: Exs Example 2: Exs Example 3: Exs P P 28. T U 29. J V R L 0 K 176 hapter Triangle Relationships

8 Student Help VISUL STRTEGY In Exs draw a sketch with side lengths or angle measures that are roughly correct, as shown on p EXMPLE lassify Triangles lassify the triangle described. a. Side lengths: 6, 8, 9 b. ngle measures: 0, 60, 70 Solution You may want to sketch the triangle. a. b Because the triangle has three sides with different lengths, the triangle is scalene Because the triangle has three angles with measures less than 90, the triangle is acute. Matching Triangles In Exercises 30 36, use the example above to match the triangle description with the most specific name. 30. Side lengths: 2 cm, 3 cm, cm. Equilateral 31. Side lengths: 3 cm, 2 cm, 3 cm B. Scalene 32. Side lengths: cm, cm, cm. Obtuse 33. ngle measures: 60, 60, 60 D. Equiangular 3. ngle measures: 30, 60, 90 E. Isosceles 3. ngle measures: 20, 1, 1 F. cute Sports 36. ngle measures: 0,, 7 G. Right Basketball The diagram shows the position and spacing of five basketball players running the triangle offense. 37. What type of triangle is formed by players, B, and? BSKETBLL The triangle offense is used by many professional teams. Players are usually spaced 1 feet to 18 feet apart from each other. This provides many options for passing so a player can make a basket. 38. What type of triangle is formed by players, D, and E? 39. What type of triangle is formed by players B, D, and E? 0. Which three players appear to form an obtuse triangle? 1. Which three players appear to form a scalene triangle? B D.1 lassifying Triangles 177 E

9 Visualize It! To help you determine the side opposite a vertex, you can draw an arrow from the vertex. F Identifying Parts of Triangles Identify which side is opposite each angle. 2. B 3. D. G E F H J E G FG& is opposite ae.. L 6. P 7. K S M N P R T Visualize It! Draw an example of the triangle. 8. obtuse scalene 9. right isosceles 0. acute scalene 1. right scalene 2. acute isosceles 3. obtuse isosceles Standardized Test Practice. Multiple hoice Which of the following terms can be used to describe a triangle with angle measures of 17, 17, and 16? acute B right obtuse D equiangular. Multiple hoice What side is opposite a? F B&* G B&* H &* J both B&* and &* B Mixed Review omplements and Supplements Find the value of each variable. (Lesson 2.3) x (6x 10) (11x 7) (x 3) 0 8x 2y Translations Find the image of the point using the translation (x, y) (x 2, y ). (Lesson 3.7) 9. (2, ) 60. (1, 3) 61. ( 1, 2) 62. (0, ) 63. (, 2) 6. (0, 0) 6. ( 6, ) 66. ( 3, 1) lgebra Skills Solving Equations Solve the equation. (Skills Review, p. 673) 67. x x 2x x x x (x 8) (x 1) 3x (3x 1) 9x hapter Triangle Relationships