Basics of Geometry PDF

1 Basics of Geometry 1.1 Points, Lines, and Planes 1.2 Measuring and Constructing Segments 1.3 Using Midpoint and Distance Formulas 1.4 Perimeter and Area in the Coordinate Plane...

1 Basics of Geometry 1.1 Points, Lines, and Planes 1.2 Measuring and Constructing Segments 1.3 Using Midpoint and Distance Formulas 1.4 Perimeter and Area in the Coordinate Plane 1.5 Measuring and Constructing Angles 1.6 Describing Pairs of Angles SEE the Big Idea Alamillo Alamil Al idge (p. illlo Bridge Brid (p. 53) 53) Soccer ((p. 49)) Shed ((p. 33)) Skateboard (p (p. 20) Sulfur S ulf ide ((p. Hexafluoride lfur H exafl fluorid p. 7) hs_geo_pe_01co.indd xx 1/19/15 8:15 AM  Maintaining Mathematical Proficiency Finding Absolute Value Example 1 Simplify ∣ −7 − 1 ∣. ∣ −7 − 1 ∣ = ∣ −7 + (−1) ∣ Add the opposite of 1. = ∣ −8 ∣ Add. =8 Find the absolute value. ∣ −7 − 1 ∣ = 8 Simplify the expression. 1. ∣ 8 − 12 ∣ 2. ∣ −6 − 5 ∣ 3. ∣ 4 + (−9) ∣ 4. ∣ 13 + (−4) ∣ 5. ∣ 6 − (−2) ∣ 6. ∣ 5 − (−1) ∣ 7. ∣ −8 − (−7) ∣ 8. ∣ 8 − 13 ∣ 9. ∣ −14 − 3 ∣ Finding the Area of a Triangle Example 2 Find the area of the triangle. 5 cm 18 cm 1 A = —2 bh Write the formula for area of a triangle. 1 = —2 (18)(5) Substitute 18 for b and 5 for h. 1 = —2 (90) Multiply 18 and 5. = 45 Multiply —12 and 90. The area of the triangle is 45 square centimeters. Find the area of the triangle. 10. 11. 12. 7 yd 22 m 16 in. 24 yd 25 in. 14 m 13. ABSTRACT REASONING Describe the possible values for x and y when ∣ x − y ∣ > 0. What does it mean when ∣ x − y ∣ = 0? Can ∣ x − y ∣ < 0? Explain your reasoning. Dynamic Solutions available at BigIdeasMath.com 1 hs_geo_pe_01co.indd 1 1/19/15 8:15 AM  Mathematical Mathematically proficient students carefully specify units of measure. Practices Specifying Units of Measure Core Concept Customary Units of Length Metric Units of Length 1 foot = 12 inches 1 centimeter = 10 millimeters 1 yard = 3 feet 1 meter = 1000 millimeters 1 mile = 5280 feet = 1760 yards 1 kilometer = 1000 meters in. 1 2 3 1 in. = 2.54 cm cm 1 2 3 4 5 6 7 8 9 Converting Units of Measure Find the area of the rectangle in square centimeters. 2 in. Round your answer to the nearest hundredth. SOLUTION 6 in. Use the formula for the area of a rectangle. Convert the units of length from customary units to metric units. Area = (Length)(Width) Formula for area of a rectangle = (6 in.)(2 in.) Substitute given length and width. [ ( 2.54 cm = (6 in.) — 1 in. ) ] [ (2 in.)( 2.541 in.cm ) ] — Multiply each dimension by the conversion factor. = (15.24 cm)(5.08 cm) Multiply. ≈ 77.42 cm2 Multiply and round to the nearest hundredth. The area of the rectangle is about 77.42 square centimeters. Monitoring Progress Find the area of the polygon using the specified units. Round your answer to the nearest hundredth. 1. triangle (square inches) 2. parallelogram (square centimeters) 2 cm 2 in. 2 cm 2.5 in. 3. The distance between two cities is 120 miles. What is the distance in kilometers? Round your answer to the nearest whole number. 2 Chapter 1 Basics of Geometry hs_geo_pe_01co.indd 2 1/19/15 8:15 AM  1.1 Points, Lines, and Planes Essential Question How can you use dynamic geometry software to visualize geometric concepts? Using Dynamic Geometry Software Work with a partner. Use dynamic geometry software to draw several points. Also, draw some lines, line segments, and rays. What is the difference between a line, a line segment, and a ray? Sample B G A F C E D Intersections of Lines and Planes Work with a partner. Q a. Describe and sketch the ways in which two lines can intersect or not intersect. Give examples of each using the lines formed by the walls, floor, and ceiling in your classroom. B P b. Describe and sketch the ways in which a line and a plane can intersect or not intersect. Give examples of each using the walls, A UNDERSTANDING floor, and ceiling in your classroom. MATHEMATICAL c. Describe and sketch the ways in which TERMS two planes can intersect or not intersect. To be proficient in math, Give examples of each using the walls, you need to understand floor, and ceiling in your classroom. definitions and previously established results. An appropriate tool, such as a software package, Exploring Dynamic Geometry Software can sometimes help. Work with a partner. Use dynamic geometry software to explore geometry. Use the software to find a term or concept that is unfamiliar to you. Then use the capabilities of the software to determine the meaning of the term or concept. Communicate Your Answer 4. How can you use dynamic geometry software to visualize geometric concepts? Section 1.1 Points, Lines, and Planes 3 hs_geo_pe_0101.indd 3 1/19/15 8:18 AM  1.1 Lesson What You Will Learn Name points, lines, and planes. Name segments and rays. Core Vocabul Vocabulary larry Sketch intersections of lines and planes. undefined terms, p. 4 Solve real-life problems involving lines and planes. point, p. 4 line, p. 4 plane, p. 4 Using Undefined Terms collinear points, p. 4 In geometry, the words point, line, and plane are undefined terms. These words do coplanar points, p. 4 not have formal definitions, but there is agreement about what they mean. defined terms, p. 5 line segment, or segment, p. 5 endpoints, p. 5 Core Concept ray, p. 5 opposite rays, p. 5 Undefined Terms: Point, Line, and Plane A intersection, p. 6 Point A point has no dimension. A dot represents a point. point A Line A line has one dimension. It is represented by a A line with two arrowheads, but it extends without end. B Through any two points, there is exactly one line. You can use any two points on a line to name it. line , line AB (AB), or line BA (BA) Plane A plane has two dimensions. It is represented by a shape that looks like a floor or a wall, but it A M extends without end. C B Through any three points not on the same line, there is exactly one plane. You can use three points that plane M, or plane ABC are not all on the same line to name a plane. Collinear points are points that lie on the same line. Coplanar points are points that lie in the same plane. Naming Points, Lines, and Planes n a. Give two other names for ⃖⃗ PQ and plane R. Q b. Name three points that are collinear. Name four V T m points that are coplanar. S P R SOLUTION a. Other names for ⃖⃗ PQ are ⃖⃗ QP and line n. Other names for plane R are plane SVT and plane PTV. b. Points S, P, and T lie on the same line, so they are collinear. Points S, P, T, and V lie in the same plane, so they are coplanar. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 1. Use the diagram in Example 1. Give two other names for ⃖⃗ ST. Name a point that is not coplanar with points Q, S, and T. 4 Chapter 1 Basics of Geometry hs_geo_pe_0101.indd 4 1/19/15 8:18 AM  Using Defined Terms In geometry, terms that can be described using known words such as point or line are called defined terms. Core Concept Defined Terms: Segment and Ray line The definitions below use line AB (written as ⃖⃗ AB) and points A and B. A B Segment The line segment AB, or segment AB, — ) consists of the endpoints A and B (written as AB segment and all points on ⃖⃗ AB that are between A and B. endpoint endpoint — can also be named BA Note that AB —. A B Ray The ray AB (written as ⃗ AB ) consists of the ray endpoint A and all points on ⃖⃗ AB that lie on the endpoint same side of A as B. A B Note that ⃗ AB and ⃗ BA are different rays. endpoint A B Opposite Rays If point C lies on ⃖⃗ AB between A and B, then ⃗ CA and ⃗ CB are opposite rays. A C B Segments and rays are collinear when they lie on the same line. So, opposite rays are collinear. Lines, segments, and rays are coplanar when they lie in the same plane. Naming Segments, Rays, and Opposite Rays — a. Give another name for GH. E G COMMON ERROR b. Name all rays with endpoint J. Which F In Example 2, ⃗ JG and ⃗ JF of these rays are opposite rays? J have a common endpoint, H but they are not collinear. So, SOLUTION they are not opposite rays. — — a. Another name for GH is HG. JE , ⃗ b. The rays with endpoint J are ⃗ JG , ⃗ JF , and ⃗ JH. The pairs of opposite rays with endpoint J are ⃗ JF , and ⃗ JE and ⃗ JG and ⃗ JH. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Use the diagram. K M P L N 2. Give another name for KL. — 3. Are ⃗ KP and ⃗ PK the same ray? Are ⃗ ⃗ the same ray? Explain. NP and NM Section 1.1 Points, Lines, and Planes 5 hs_geo_pe_0101.indd 5 1/19/15 8:18 AM  Sketching Intersections Two or more geometric figures intersect when they have one or more points in common. The intersection of the figures is the set of points the figures have in common. Some examples of intersections are shown below. m A n q The intersection of two different lines is a point. The intersection of two different planes is a line. Sketching Intersections of Lines and Planes a. Sketch a plane and a line that is in the plane. b. Sketch a plane and a line that does not intersect the plane. c. Sketch a plane and a line that intersects the plane at a point. SOLUTION a. b. c. Sketching Intersections of Planes Sketch two planes that intersect in a line. SOLUTION Step 1 Draw a vertical plane. Shade the plane. Step 2 Draw a second plane that is horizontal. Shade this plane a different color. Use dashed lines to show where one plane is hidden. Step 3 Draw the line of intersection. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 4. Sketch two different lines that intersect a plane B at the same point. Use the diagram. k P Q 5. Name the intersection of ⃖⃗ PQ and line k. M A 6. Name the intersection of plane A and plane B. 7. Name the intersection of line k and plane A. 6 Chapter 1 Basics of Geometry hs_geo_pe_0101.indd 6 1/19/15 8:18 AM  Solving Real-Life Problems Modeling with Mathematics The diagram shows a molecule of sulfur hexafluoride, the most potent greenhouse gas in the world. Name two different planes that contain line r. p A q D B E r G F C SOLUTION Electric utilities use sulfur hexafluoride as an insulator. Leaks in electrical 1. Understand the Problem In the diagram, you are given three lines, p, q, and r, equipment contribute to the release of that intersect at point B. You need to name two different planes that contain line r. sulfur hexafluoride into the atmosphere. 2. Make a Plan The planes should contain two points on line r and one point not on line r. 3. Solve the Problem Points D and F are on line r. Point E does not lie on line r. So, plane DEF contains line r. Another point that does not lie on line r is C. So, plane CDF contains line r. Note that you cannot form a plane through points D, B, and F. By definition, three points that do not lie on the same line form a plane. Points D, B, and F are collinear, so they do not form a plane. 4. Look Back The question asks for two different planes. You need to check whether plane DEF and plane CDF are two unique planes or the same plane named differently. Because point C does not lie on plane DEF, plane DEF and plane CDF are different planes. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Use the diagram that shows a molecule of phosphorus pentachloride. s G J K H L I 8. Name two different planes that contain line s. 9. Name three different planes that contain point K. 10. Name two different planes that contain ⃗ HJ. Section 1.1 Points, Lines, and Planes 7 hs_geo_pe_0101.indd 7 1/19/15 8:18 AM  1.1 Exercises Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check 1. WRITING Compare collinear points and coplanar points. 2. WHICH ONE DOESN’T BELONG? Which term does not belong with the other three? Explain your reasoning. — AB plane CDE ⃖⃗ FG ⃗ HI Monitoring Progress and Modeling with Mathematics In Exercises 3–6, use the diagram. In Exercises 11–16, use the diagram. (See Example 2.) B S t C B A E s D T A E C 3. Name four points. D 4. Name two lines. 11. What is another name for BD ? — 5. Name the plane that contains points A, B, and C. 12. What is another name for AC ? — 6. Name the plane that contains points A, D, and E. 13. What is another name for ray ⃗ AE? In Exercises 7–10, use the diagram. (See Example 1.) 14. Name all rays with endpoint E. g 15. Name two pairs of opposite rays. W 16. Name one pair of rays that are not opposite rays. f S Q In Exercises 17–24, sketch the figure described. R T V (See Examples 3 and 4.) 17. plane P and lineℓ intersecting at one point 18. plane K and line m intersecting at all points on line m Give two other names for ⃖⃗ 19. ⃗ AB and ⃖⃗ 7. WQ. AC 8. Give another name for plane V. ⃗ and NX 20. MN ⃗ 9. Name three points that are collinear. Then name 21. plane M and ⃗ NB intersecting at B a fourth point that is not collinear with these three points. 22. plane M and ⃗ NB intersecting at A 23. plane A and plane B not intersecting 10. Name a point that is not coplanar with R, S, and T. 24. plane C and plane D intersecting at ⃖⃗ XY 8 Chapter 1 Basics of Geometry hs_geo_pe_0101.indd 8 1/19/15 8:18 AM  ERROR ANALYSIS In Exercises 25 and 26, describe In Exercises 35–38, name the geometric term modeled and correct the error in naming opposite rays in by the object. the diagram. 35. C A B X D Y E 36. ✗ 25. ⃗ AD and ⃗ AC are opposite rays. ✗ 26. 37. 38. — and YE YC — are opposite rays. In Exercises 27–34, use the diagram. B I C A D In Exercises 39–44, use the diagram to name all the points that are not coplanar with the given points. F 39. N, K, and L K G L 40. P, Q, and N E J H N 41. P, Q, and R M 27. Name a point that is collinear with points E and H. 42. R, K, and N R S 28. Name a point that is collinear with points B and I. 43. P, S, and K 29. Name a point that is not collinear with points E 44. Q, K, and L Q P and H. 45. CRITICAL THINKING Given two points on a line and 30. Name a point that is not collinear with points B and I. a third point not on the line, is it possible to draw a plane that includes the line and the third point? 31. Name a point that is coplanar with points D, A, and B. Explain your reasoning. 32. Name a point that is coplanar with points C, G, and F. 46. CRITICAL THINKING Is it possible for one point to be in two different planes? Explain your reasoning. 33. Name the intersection of plane AEH and plane FBE. 34. Name the intersection of plane BGF and plane HDG. Section 1.1 Points, Lines, and Planes 9 hs_geo_pe_0101.indd 9 1/19/15 8:18 AM  47. REASONING Explain why a four-legged chair may 53. x ≥ 5 or x ≤ −2 54. ∣x∣ ≤ 0 rock from side to side even if the floor is level. Would a three-legged chair on the same level floor rock from 55. MODELING WITH MATHEMATICS Use the diagram. side to side? Why or why not? J L 48. THOUGHT PROVOKING You are designing the living room of an apartment. Counting the floor, walls, and K ceiling, you want the design to contain at least eight different planes. Draw a diagram of your design. P Label each plane in your design. Q N M 49. LOOKING FOR STRUCTURE Two coplanar intersecting lines will always intersect at one point. What is the greatest number of intersection points that exist if you draw four coplanar lines? Explain. 50. HOW DO YOU SEE IT? You and your friend walk in opposite directions, forming opposite rays. You a. Name two points that are collinear with P. were originally on the corner of Apple Avenue and b. Name two planes that contain J. Cherry Court. c. Name all the points that are in more than one plane. N W E Apple Ave. CRITICAL THINKING In Exercises 56–63, complete the S Rd. statement with always, sometimes, or never. Explain Cherry Ct. Rose your reasoning. 56. A line ____________ has endpoints. Daisy Dr. 57. A line and a point ____________ intersect. 58. A plane and a point ____________ intersect. a. Name two possibilities of the road and direction 59. Two planes ____________ intersect in a line. you and your friend may have traveled. 60. Two points ____________ determine a line. b. Your friend claims he went north on Cherry Court, and you went east on Apple Avenue. 61. Any three points ____________ determine a plane. Make an argument as to why you know this could not have happened. 62. Any three points not on the same line ____________ determine a plane. MATHEMATICAL CONNECTIONS In Exercises 51–54, 63. Two lines that are not parallel ___________ intersect. graph the inequality on a number line. Tell whether the 64. ABSTRACT REASONING Is it possible for three planes graph is a segment, a ray or rays, a point, or a line. to never intersect? intersect in one line? intersect in 51. x ≤ 3 52. −7 ≤ x ≤ 4 one point? Sketch the possible situations. Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Find the absolute value. (Skills Review Handbook) 65. ∣6 + 2∣ 66. ∣3 − 9∣ 67. ∣ −8 − 2 ∣ 68. ∣ 7 − 11 ∣ Solve the equation. (Skills Review Handbook) 69. 18 + x = 43 70. 36 + x = 20 71. x − 15 = 7 72. x − 23 = 19 10 Chapter 1 Basics of Geometry hs_geo_pe_0101.indd 10 1/19/15 8:18 AM  1.2 Measuring and Constructing Segments Essential Question How can you measure and construct a line segment? Measuring Line Segments Using Nonstandard Units 12 Work with a partner. 10 11 3 2 1 CM 4 9 5 6 a. Draw a line segment that has a length 7 8 9 8 7 10 11 12 6 of 6 inches. 13 14 5 15 16 17 4 19 18 b. Use a standard-sized paper clip to 3 20 21 22 2 24 23 1 25 measure the length of the line segment. INC H 30 29 28 27 26 Explain how you measured the line MAKING SENSE segment in “paper clips.” OF PROBLEMS To be proficient in math, you need to explain to c. Write conversion factors from paper clips ps yourself the meaning of to inches and vice versa. a problem and look for entry points to its solution. 1 paper clip = in. 1 in. = paper clip d. A straightedge is a tool that you can use to draw a straight line. An 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. Measuring Line Segments Using Nonstandard Units Work with a partner. a. Fold a 3-inch by 5-inch index card fo on one of its diagonals. ld 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. Measuring Heights Using Nonstandard Units Work with a partner. Consider a unit of length that is equal to the length of the diagonal you found in Exploration 2. Call this length “1 diag.” How tall are you in diags? Explain how you obtained your answer. Communicate Your Answer 4. How can you measure and construct a line segment? Section 1.2 Measuring and Constructing Segments 11 hs_geo_pe_0102.indd 11 1/19/15 8:18 AM  1.2 Lesson What You Will Learn Use the Ruler Postulate. Copy segments and compare segments for congruence. Core Vocabul Vocabulary larry Use the Segment Addition Postulate. postulate, p. 12 axiom, p. 12 Using the Ruler Postulate coordinate, p. 12 distance, p. 12 In geometry, a rule that is accepted without proof is called a postulate or an axiom. A rule that can be proved is called a theorem, as you will see later. Postulate 1.1 shows construction, p. 13 how to find the distance between two points on a line. congruent segments, p. 13 between, p. 14 Postulate names of points Postulate 1.1 Ruler Postulate A B The points on a line can be matched one to one with the real numbers. The real number x1 x2 that corresponds to a point is the coordinate of the point. coordinates of points The distance between points A and B, written A AB B as AB, is the absolute value of the difference of the coordinates of A and B. x1 x2 AB = x2 − x1 Using the Ruler Postulate — Measure the length of ST to the nearest tenth of a centimeter. S T SOLUTION Align one mark of a metric ruler with S. Then estimate the coordinate of T. For example, when you align S with 2, T appears to align with 5.4. S T cm 1 2 3 4 5 6 ST = ∣ 5.4 − 2 ∣ = 3.4 Ruler Postulate — is about 3.4 centimeters. So, the length of ST Monitoring Progress Help in English and Spanish at BigIdeasMath.com Use a ruler to measure the length of the segment to the nearest —18 inch. 1. 2. M N P Q 3. 4. U V W X 12 Chapter 1 Basics of Geometry hs_geo_pe_0102.indd 12 1/19/15 8:19 AM  Constructing and Comparing Congruent Segments A construction is a geometric drawing that uses a limited set of tools, usually a compass and straightedge. Copying a Segment Use a compass and straightedge to construct a line segment —. that has the same length as AB A B SOLUTION Step 1 Step 2 Step 3 A B A B A B C C C D Draw a segment Use a straightedge Measure length Set your Copy length Place the compass at —. to draw a segment longer than AB —. compass at the length of AB C. Mark point D on the new segment. Label point C on the new segment. — has the same length as AB So, CD —. Core Concept Congruent Segments READING Line segments that have the same length are called congruent segments. You In the diagram, the — is equal to the length of CD can say “the length of AB —,” or you can say “AB — is red tick marks indicate — congruent to CD.” The symbol ≅ means “is congruent to.” — AB ≅ — CD. When there Lengths are equal. Segments are congruent. is more than one pair of A B AB = CD — ≅ CD AB — congruent segments, use multiple tick marks. C D “is equal to” “is congruent to” Comparing Segments for Congruence Plot J(−3, 4), K(2, 4), L(1, 3), and M(1, −2) in a coordinate plane. Then determine — and LM whether JK — are congruent. SOLUTION y Plot the points, as shown. To find the length of a horizontal segment, find the absolute J(−3, 4) K(2, 4) value of the difference of the x-coordinates of the endpoints. L(1, 3) JK = ∣ 2 − (−3) ∣ = 5 Ruler Postulate 2 To find the length of a vertical segment, find the absolute value of the difference of the y-coordinates of the endpoints. −4 −2 2 4 x LM = ∣ −2 − 3 ∣ = 5 Ruler Postulate −2 M(1, −2) — — have the same length. So, JK JK and LM — ≅ LM —. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 5. Plot A(−2, 4), B(3, 4), C(0, 2), and D(0, −2) in a coordinate plane. Then — and CD determine whether AB — are congruent. Section 1.2 Measuring and Constructing Segments 13 hs_geo_pe_0102.indd 13 1/19/15 8:19 AM  Using the Segment Addition Postulate When three points are collinear, you can say that one point is between the other two. A D E B C F Point B is between Point E is not between points A and C. points D and F. Postulate Postulate 1.2 Segment Addition Postulate If B is between A and C, then AB + BC = AC. AC If AB + BC = AC, then B is between A and C. A B C AB BC Using the Segment Addition Postulate a. Find DF. D 23 E 35 F b. Find GH. 36 F 21 G H SOLUTION a. Use the Segment Addition Postulate to write an equation. Then solve the equation to find DF. DF = DE + EF Segment Addition Postulate DF = 23 + 35 Substitute 23 for DE and 35 for EF. DF = 58 Add. b. Use the Segment Addition Postulate to write an equation. Then solve the equation to find GH. FH = FG + GH Segment Addition Postulate 36 = 21 + GH Substitute 36 for FH and 21 for FG. 15 = GH Subtract 21 from each side. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Use the diagram at the right. 23 50 6. Use the Segment Addition Postulate to find XZ. X Y Z 7. In the diagram, WY = 30. Can you use the W Segment Addition Postulate to find the distance between points W and Z ? Explain your reasoning. 144 8. Use the diagram at the left to find KL. J 37 K L 14 Chapter 1 Basics of Geometry hs_geo_pe_0102.indd 14 1/19/15 8:19 AM  Using the Segment Addition Postulate The cities shown on the map lie approximately in a straight line. Find the distance from Tulsa, Oklahoma, to St. Louis, Missouri. S St. Louis 738 mi T Tulsa 377 mi L Lubbock SOLUTION 1. Understand the Problem You are given the distance from Lubbock to St. Louis and the distance from Lubbock to Tulsa. You need to find the distance from Tulsa to St. Louis. 2. Make a Plan Use the Segment Addition Postulate to find the distance from Tulsa to St. Louis. 3. Solve the Problem Use the Segment Addition Postulate to write an equation. Then solve the equation to find TS. LS = LT + TS Segment Addition Postulate 738 = 377 + TS Substitute 738 for LS and 377 for LT. 361 = TS Subtract 377 from each side. So, the distance from Tulsa to St. Louis is about 361 miles. 4. Look Back Does the answer make sense in the context of the problem? The distance from Lubbock to St. Louis is 738 miles. By the Segment Addition Postulate, the distance from Lubbock to Tulsa plus the distance from Tulsa to St. Louis should equal 738 miles. 377 + 361 = 738 ✓ Monitoring Progress Help in English and Spanish at BigIdeasMath.com 9. The cities shown on the map lie approximately in a straight line. Find the distance from Albuquerque, New Mexico, to Provo, Utah. Provo P 680 mi Albuquerque A 231 mi Carlsbad C Section 1.2 Measuring and Constructing Segments 15 hs_geo_pe_0102.indd 15 1/19/15 8:19 AM  1.2 Exercises Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check — 1. WRITING Explain how XY and XY are different. 2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers. 7 B 3 C A Find AC + CB. Find BC − AC. Find AB. Find CA + BC. Monitoring Progress and Modeling with Mathematics In Exercises 3 –6, use a ruler to measure the length of In Exercises 15–22, find FH. (See Example 3.) the segment to the nearest tenth of a centimeter. 15. 8 (See Example 1.) F 3. G 14 4. H 5. 16. 7 H 19 G 6. F CONSTRUCTION In Exercises 7 and 8, use a compass and straightedge to construct a copy of the segment. 17. 12 H 7. Copy the segment in Exercise 3. 11 G 8. Copy the segment in Exercise 4. F In Exercises 9–14, plot the points in a coordinate 18. plane. Then determine whether — AB and — 4 CD are F congruent. (See Example 2.) G 15 9. A(−4, 5), B(−4, 8), C(2, −3), D(2, 0) H 10. A(6, −1), B(1, −1), C(2, −3), D(4, −3) 19. 11. A(8, 3), B(−1, 3), C(5, 10), D(5, 3) 37 G H 13 12. A(6, −8), B(6, 1), C(7, −2), D(−2, −2) F 13. A(−5, 6), B(−5, −1), C(−4, 3), D(3, 3) 14. A(10, −4), B(3, −4), C(−1, 2), D(−1, 5) 16 Chapter 1 Basics of Geometry hs_geo_pe_0102.indd 16 1/19/15 8:19 AM  20. 22 26. MODELING WITH MATHEMATICS In 2003, a remote- controlled model airplane became the first ever to fly F H 15 G nonstop across the Atlantic Ocean. The map shows the airplane’s position at three different points during its 21. flight. Point A represents Cape Spear, Newfoundland, F point B represents the approximate position after 42 1 day, and point C represents Mannin Bay, Ireland. The airplane left from Cape Spear and landed in H Mannin Bay. (See Example 4.) 22 G 22. Europe North C G America B A 601 mi 1282 mi 53 40 Atlantic Ocean H F a. Find the total distance the model airplane flew. b. The model airplane’s flight lasted nearly 38 hours. ERROR ANALYSIS In Exercises 23 and 24, describe and correct the error in finding the length of — Estimate the airplane’s average speed in AB. miles per hour. A B 27. USING STRUCTURE Determine whether the statements are true or false. Explain your reasoning. cm 1 2 3 4 5 6 C B ✗ 23. AB = 1 − 4.5 = −3.5 A D E H F ✗ 24. a. B is between A and C. AB = ∣ 1 + 4.5 ∣ = 5.5 b. C is between B and E. c. D is between A and H. 25. ATTENDING TO PRECISION The diagram shows an d. E is between C and F. insect called a walking stick. Use the ruler to estimate the length of the abdomen and the length of the 28. MATHEMATICAL CONNECTIONS Write an expression thorax to the nearest —14 inch. How much longer is the for the length of the segment. walking stick’s abdomen than its thorax? How many — a. AC times longer is its abdomen than its thorax? abdomen thorax A x+2 B 7x − 3 C — b. QR 13y + 25 P 8y + 5 Q R INCH 1 2 3 4 5 6 7 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Section 1.2 Measuring and Constructing Segments 17 hs_geo_pe_0102.indd 17 1/19/15 8:19 AM  29. MATHEMATICAL CONNECTIONS Point S is between 33. REASONING You travel from City X to City Y. You —. Use the information to write an points R and T on RT know that the round-trip distance is 647 miles. City Z, a equation in terms of x. Then solve the equation and city you pass on the way, is 27 miles from City X. Find find RS, ST, and RT. the distance from City Z to City Y. Justify your answer. a. RS = 2x + 10 b. RS = 3x − 16 ST = x − 4 ST = 4x − 8 34. HOW DO YOU SEE IT? The bar graph shows the win-loss record for a lacrosse team over a period of RT = 21 RT = 60 three years. Explain how you can apply the Ruler c. RS = 2x − 8 d. RS = 4x − 9 Postulate (Post. 1.1) and the Segment Addition ST = 11 ST = 19 Postulate (Post. 1.2) when interpreting a stacked bar graph like the one shown. RT = x + 10 RT = 8x − 14 Win-Loss Record 30. THOUGHT PROVOKING Is it possible to design a table where no two legs have the same length? Assume Wins Losses that the endpoints of the legs must all lie in the same Year 1 plane. Include a diagram as part of your answer. Year 2 Year 3 31. MODELING WITH MATHEMATICS You have to walk 0 2 4 6 8 10 12 from Room 103 to Room 117. Number of games 103 107 113 117 86 ft 35. ABSTRACT REASONING The points (a, b) and (c, b) 22 ft form a segment, and the points (d, e) and (d, f ) form a segment. Create an equation assuming the segments 101 105 109 111 115 119 121 are congruent. Are there any letters not used in the equation? Explain. a. How many feet do you travel from Room 103 36. MATHEMATICAL CONNECTIONS In the diagram, to Room 117? — ≅ BC AB —, AC— ≅ CD —, and AD = 12. Find the lengths of all segments in the diagram. Suppose you b. You can walk 4.4 feet per second. How many choose one of the segments at random. What is the minutes will it take you to get to Room 117? probability that the measure of the segment is greater c. Why might it take you longer than the time in than 3? Explain your reasoning. part (b)? D 32. MAKING AN ARGUMENT Your friend and your cousin C B discuss measuring with a ruler. Your friend says that A you must always line up objects at the zero on a ruler. Your cousin says it does not matter. Decide who is 37. CRITICAL THINKING Is it possible to use the Segment correct and explain your reasoning. Addition Postulate (Post. 1.2) to show FB > CB or that AC > DB? Explain your reasoning. A D F C B Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Simplify. (Skills Review Handbook) −4 + 6 — — 7+6 38. — 39. √ 20 + 5 40. √ 25 + 9 41. — 2 2 Solve the equation. (Skills Review Handbook) 3+y −5 + x 42. 5x + 7 = 9x − 17 43. — = 6 44. — = −9 45. −6x − 13 = −x − 23 2 2 18 Chapter 1 Basics of Geometry hs_geo_pe_0102.indd 18 1/19/15 8:19 AM  1.3 Using Midpoint and Distance Formulas Essential Question How can you find the midpoint and length of a line segment in a coordinate plane? Finding the Midpoint of a Line Segment Work with a partner. Use centimeter graph paper. a. Graph AB—, where the points A and B are as shown. b. Explain how to bisect AB—, 4 — that is, to divide AB into two A(3, 4) congruent line