M 1. TEXS ESSENTIL KNOWLEDE ND SKILLS...5. Preparing for.5. USIN PROLEM-SOLVIN STRTEIES To be proficient in math, you need to explain to yourself the meaning of a problem and look for entry points to its solution. Measuring and onstructing Segments Essential Question ow can you measure and construct a line segment? Work with a partner. a. Draw a line segment that has a length of 6 inches. b. Use a standard-sized paper clip to measure the length of the line segment. Explain how you measured the line segment in paper clips. Measuring Line Segments Using Nonstandard Units c. Write conversion factors from paper clips to inches and vice versa. 1 paper clip = in. 1 in. = paper clip d. straightedge is a tool that you can use to draw a straight line. n example of a straightedge is a ruler. Use only a pencil, straightedge, paper clip, and paper to draw another line segment that is 6 inches long. Explain your process. 0 1 3 4 5 6 7 8 9 3 0 IN 1 3 4 5 6 7 8 9 1 0 1 1 1 1 3 1 4 1 5 1 6 1 7 1 8 1 9 0 1 3 4 5 6 7 8 9 10 11 1 Measuring Line Segments Using Nonstandard Units Work with a partner. a. old a 3-inch by 5-inch index card on one of its diagonals. 3 in. b. Use the Pythagorean Theorem to algebraically determine the length of the diagonal in inches. Use a ruler to check your answer. 5 in. c. Measure the length and width of the index card in paper clips. d. Use the Pythagorean Theorem to algebraically determine the length of the diagonal in paper clips. Then check your answer by measuring the length of the diagonal in paper clips. Does the Pythagorean Theorem work for any unit of measure? Justify your answer. fold Measuring eights Using Nonstandard Units Work with a partner. onsider a unit of length that is equal to the length of the diagonal you found in Exploration. all this length 1 diag. ow tall are you in diags? Explain how you obtained your answer. ommunicate Your nswer 4. ow can you measure and construct a line segment? Section 1. Measuring and onstructing Segments 11
1. Lesson ore Vocabulary postulate, p. 1 axiom, p. 1 coordinate, p. 1 distance, p. 1 between, p. 13 construction, p. 15 congruent segments, p. 15 What You Will Learn Use the Ruler Postulate. Use the Segment ddition Postulate. Use the Distance ormula. opy segments and compare segments for congruence. Using the Ruler Postulate In geometry, a rule that is accepted without proof is called a postulate or an axiom. rule that can be proved is called a theorem, as you will see later. Postulate 1.1 shows how to find the distance between two points on a line. Postulate Postulate 1.1 Ruler Postulate The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point. names of points x 1 x coordinates of points The distance between points and, written as, is the absolute value of the difference of the coordinates of and. x 1 x = x x 1 Using the Ruler Postulate ind the length of E. E 4 3 1 0 1 3 4 5 6 y the Ruler Postulate, the coordinate of is 1 and the coordinate of E is 5. E = 5 ( 1 Definition of distance = 5 + 1 Simplify. = 6 dd. = 6 ind the absolute value. So, the length of E is 6. Monitoring Progress ind the length of the segment. J elp in English and Spanish at igideasmath.com K L M 6 5 4 3 1 0 1 3 4 1. MN. LN 3. KJ 4. JL N 1 hapter 1 asics of eometry
Using the Segment ddition Postulate When three points are collinear, you can say that one point is between the other two. D E Point is between points and. Point E is not between points D and. Postulate Postulate 1. Segment ddition Postulate If is between and, then + =. If + =, then is between and. Using the Segment ddition Postulate a. ind D. b. ind. D 3 E 35 36 1 a. Use the Segment ddition Postulate to write an equation. Then solve the equation to find D. D = DE + E D = 3 + 35 D = 58 Segment ddition Postulate Substitute 3 for DE and 35 for E. dd. b. Use the Segment ddition Postulate to write an equation. Then solve the equation to find. = + Segment ddition Postulate 36 = 1 + Substitute 36 for and 1 for. 15 = Subtract 1 from each side. 144 J 37 K L Monitoring Progress Use the diagram at the right. 5. Use the Segment ddition Postulate to find XZ. 6. In the diagram, WY = 30. an you use the Segment ddition Postulate to find the distance between points W and Z? Explain your reasoning. 7. Use the diagram at the left to find KL. elp in English and Spanish at igideasmath.com X 3 W Y 50 Z Section 1. Measuring and onstructing Segments 13
REDIN The red mark at the corner of the triangle that makes a right angle indicates a right triangle. Using the Distance ormula You can use the Distance ormula to find the distance between two points in a coordinate plane. ore oncept The Distance ormula If (x 1, y 1 and (x, y are points in a coordinate plane, then the distance between and is = (x x 1 + (y y 1. The Distance ormula is related to the Pythagorean Theorem, which you will see again when you work with right triangles. y (x 1, y 1 (x, y x x 1 y y 1 (x, y 1 x Distance ormula ( = (x x 1 + (y y 1 Pythagorean Theorem c = a + b y (x 1, y 1 x x 1 (x, y y y 1 (x, y 1 x c a b Using the Distance ormula Your school is 4 miles east and 1 mile south of your apartment. recycling center, where your class is going on a field trip, is miles east and 3 miles north of your apartment. Estimate the distance between the recycling center and your school. REDIN The symbol means is approximately equal to. You can model the situation using a coordinate plane with your apartment at the origin (0, 0. The coordinates of the recycling center and the school are R(, 3 and S(4, 1, respectively. Use the Distance ormula. Let (x 1, y 1 = (, 3 and (x, y = (4, 1. RS = (x x 1 + (y y 1 Distance ormula = (4 + ( 1 3 Substitute. = + ( 4 Subtract. = 4 + 16 Evaluate powers. = 0 dd. 4.5 Use a calculator. W 4 N S R(, 3 S(4, 1 E 14 hapter 1 asics of eometry So, the distance between the recycling center and your school is about 4.5 miles. Monitoring Progress elp in English and Spanish at igideasmath.com 8. In Example 3, a park is 3 miles east and 4 miles south of your apartment. ind the distance between the park and your school.
onstructing and omparing ongruent Segments construction is a geometric drawing that uses a limited set of tools, usually a compass and straightedge. opying a Segment Use a compass and straightedge to construct a line segment that has the same length as. Step 1 Step Step 3 D Draw a segment Use a straightedge to draw a segment longer than. Label point on the new segment. Measure length Set your compass at the length of. opy length Place the compass at. Mark point D on the new segment. So, D has the same length as. REDIN In the diagram, the red tick marks indicate D. When there is more than one pair of congruent segments, use multiple tick marks. ore oncept ongruent Segments Line segments that have the same length are called congruent segments. You can say the length of is equal to the length of D, or you can say is congruent to D. The symbol means is congruent to. Lengths are equal. Segments are congruent. = D D D is equal to is congruent to omparing Segments for ongruence J( 3, 4 4 L( 1, 3 4 y K(1, 1 M(3, 0 4 x Plot J( 3, 4, K(1, 1, L( 1, 3, and M(3, 0 in a coordinate plane. Then determine whether JK and LM are congruent. Plot the points, as shown. Use the Distance ormula to find the length of each segment. JK = [1 ( 3] + (1 4 = 4 + ( 3 = 5 = 5 LM = [3 ( 1] + [0 ( 3] = 4 + 3 = 5 = 5 JK and LM have the same length. So, JK LM. Monitoring Progress elp in English and Spanish at igideasmath.com 9. Plot (, 4, (, 3, (, 3, and D(1, 1 in a coordinate plane. Then determine whether and D are congruent. Section 1. Measuring and onstructing Segments 15
1. Exercises Dynamic Solutions available at igideasmath.com Vocabulary and ore oncept heck 1. WRITIN Explain how XY and XY are different.. DIERENT WORDS, SME QUESTION Which is different? ind both answers. 3 7 ind +. ind. ind. ind +. Monitoring Progress and Modeling with Mathematics In Exercises 3 10, find the length of the segment. (See Example 1. D E 14. 4 15 4 3 1 0 1 3 4 3. D 4. E 5. 6. D 7. 8. D 9. E 10. In Exercises 11 18, find. (See Example. 11. 1. 8 19 14 7 15. 16. 17. 37 15 4 13 18. 13. 11 1 53 40 16 hapter 1 asics of eometry
In Exercises 19 6, find the distance between the two points. (See Example 3. 19. (13, and (7, 10 0. ( 6, 5 and D( 3, 1 1. E(3, 7 and (6, 5. ( 5, 4 and (, 6 3. J( 8, 0 and K(1, 4 4. L(7, 1 and M(, 4 5. R(0, 1 and S(6, 3.5 6. T(13, 1.6 and V(5.4, 3.7 ONSTRUTION In Exercises 7 and 8, use a compass and straightedge to construct a copy of the segment. 7. 8. OMPRIN SEMENTS In Exercises 9 and 30, plot the points in a coordinate plane. Then determine whether and D are congruent. (See Example 4. 9. (0,, ( 3, 8, (,, D(0, 4 30. (1, 4, (5, 1, ( 3, 1, D(1, 6 31. MODELIN WIT MTEMTIS In 003, a remotecontrolled model airplane became the first ever to fly nonstop across the tlantic Ocean. The map shows the airplane s position at three different points during its flight. Point represents ape Spear, Newfoundland, point represents the approximate position after 1 day, and point represents Mannin ay, Ireland. The airplane left from ape Spear and landed in Mannin ay. North merica 18 mi tlantic Ocean 601 mi Europe a. ind the total distance the model airplane flew. b. The model airplane s flight lasted nearly 38 hours. Estimate the airplane s average speed in miles per hour. 3. USIN STRUTURE Determine whether the statements are true or false. Explain your reasoning. a. is between and. b. is between and E. c. D is between and. d. E is between and. D E 33. ERROR NLYSIS Describe and correct the error in finding the length of. 1 3 4 5 = 1 + 4.5 = 5.5 34. ERROR NLYSIS Describe and correct the error in finding the distance between (6, and (1, 4. = (6 1 + [ ( 4] = 5 + 6 = 5 + 36 = 61 35. MODELIN WIT MTEMTIS The figure shows the position of three players during part of a water polo match. Player throws the ball to Player, who then throws the ball to Player. Distance (m 16 1 8 4 y Player (8, 4 Player Player (18, 7 (4, 14 0 0 4 8 1 16 0 4 8 x Distance (m a. ow far did Player throw the ball? Player? b. ow far would Player have to throw the ball to throw it directly to Player? Section 1. Measuring and onstructing Segments 17
36. MODELIN WIT MTEMTIS Your school is 0 blocks east and 1 blocks south of your house. The mall is 10 blocks north and 7 blocks west of your house. You plan on going to the mall right after school. ind the distance between your school and the mall assuming there is a road directly connecting the school and the mall. One block is 0.1 mile. 37. MTEMTIL ONNETIONS Point S is between points R and T on RT. Use the information to write an equation in terms of x. Then solve the equation and find RS, ST, and RT. a. RS = x + 10 b. RS = 3x 16 ST = x 4 ST = 4x 8 RT = 1 RT = 60 c. RS = x 8 d. RS = 4x 9 ST = 11 ST = 19 RT = x + 10 RT = 8x 14 40. MKIN N RUMENT Your friend claims there is an easier way to find the length of a segment than the Distance ormula when the x-coordinates of the endpoints are equal. e claims all you have to do is subtract the y-coordinates. Do you agree with his statement? Explain your reasoning. 41. RESONIN You travel from ity X to ity Y. You know that the round-trip distance is 647 miles. ity Z, a city you pass on the way, is 7 miles from ity X. ind the distance from ity Z to ity Y. Justify your answer. 4. OW DO YOU SEE IT? The bar graph shows the win-loss record for a lacrosse team over a period of three years. Explain how you can apply the Ruler Postulate (Post. 1.1 and the Segment ddition Postulate (Post. 1. when interpreting a stacked bar graph like the one shown. Win-Loss Record 38. TOUT PROVOKIN Is it possible to design a table where no two legs have the same length? ssume that the endpoints of the legs must all lie in the same plane. Include a diagram as part of your answer. Year 1 Year Year 3 Wins Losses 39. MODELIN WIT MTEMTIS You have to walk from Room 103 to Room 117. 0 4 6 8 10 1 Number of games 101 103 107 113 117 86 ft ft 105 109 111 115 119 11 43. MTEMTIL ONNETIONS In the diagram,, D, and D = 1. ind the lengths of all segments in the diagram. Suppose you choose one of the segments at random. What is the probability that the measure of the segment is greater than 3? Explain your reasoning. a. ow many feet do you travel from Room 103 to Room 117? b. You can walk 4.4 feet per second. ow many minutes will it take you to get to Room 117? c. Why might it take you longer than the time in part (b? Maintaining Mathematical Proficiency Simplify. (Skills Review andbook 45. 4 + 6 46. 3 + 9 Solve the equation. (Skills Review andbook 49. 5x + 7 = 9x 17 50. 47. 3 + y = 6 51. 44. RITIL TINKIN Is it possible to use the Segment ddition Postulate (Post. 1. to show > or that > D? Explain your reasoning. 8 + ( 4 D Reviewing what you learned in previous grades and lessons 48. 7 + 6 5 + x = 9 5. 6x 13 = x 3 D 18 hapter 1 asics of eometry