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Chapter 1: Tools of Geometry Tools of Geometry Then You graphed points on the coordinate plane and evaluated mathematical expressions. Now In Chapter 1, you will: Find distances between points and midpoints of line segments. Identify angle relationships. Find perimeters, areas, surface areas, and volumes. KY Program of Studies HS-G-S-SR6 Students will analyze and apply spatial relationships among points, lines and planes. HS-G-S-FS4 Students will perform constructions. Why? MAPS Geometric figures and terms can be used to represent and describe real-world situations. On a map, locations of cities can be represented by points, highways or streets by lines, and national parks by polygons that have both perimeter and area. The map itself is representative of a plane. Animation glencoe.com 2 Chapter 1 Tools of Geometry Get Ready for Chapter 1 Diagnose Readiness You have two options for checking Prerequisite Skills. Text Option Take the Quick Check below. Refer to the Quick Review for help. QuickCheck QuickReview Graph and label each point in the coordinate EXAMPLE 1 (Lesson 0-7) plane. Graph and label the point Q(−3, 4) in the 1. W(5, 2) 2. X(0, 6) coordinate plane. 3. Y(-3, -1) 4. Z(4, 2) Start at the origin. y Since the x-coordinate 2 -3, 4 abcdef g h 5. GAMES Carolina is is negative, move 3 using the diagram to 8 8 7 7 units to the left. Then record her chess 6 6 move 4 units up since moves. She moves 5 5 0 the y-coordinate is x her knight 2 spaces 4 4 3 3 positive. Draw a dot up and 1 space to the 2 2 and label it Q. left from f3. What is 1 1 the location of the abcdef g h knight after Carolina completes her turn? Find each sum or difference. (Prerequisite Skill) EXAMPLE 2 2 5 1 3 1 3 6. _ + _ 7. 2 _ + 4 _ Find 3 _ + 2 _ . 3 6 18 4 6 4 13 5 3 7 _1 _3 _19 _11 8. _ - _ 9. 14 _ - 9 _ 3 + 2 = + Write as improper fractions. 18 9 5 15 6 4 6 4 19 2 11 3 _1 = _ _ + _ _ The LCD is 12. 10. FOOD Alvin ate of a pizza for dinner and 6 (2) 4 (3) 1 3 took _ of it for lunch the next day. How 38 33 6 = _ + _ Multiply. much of the pizza does he have left? 12 12 71 11 = _ or 5 _ Simplify. 12 12 Evaluate each expression. (Lesson 0-4) EXAMPLE 3 11. (-4 - 5 ) 2 12. (6 - 10 ) 2 Evaluate the expression [-2 - (-7) ] 2 + (1 - 8 ) 2 . 13. (8 - 5 ) 2 + [9 - (-3) ] 2 Follow the order of operations. [-2 - (-7) ] 2 + (1 - 8 ) 2 Solve each equation. (Lesson 0-5) = 5 2 + (-7 ) 2 Subtract. 14. 6x + 5 + 2x - 11 = 90 = 25 + 49 5 2 = 25, (-7 ) 2 = 49 15. 8x - 7 = 53 - 2x = 74 Add. Online Option KY Math Online Take a self-check Chapter Readiness Quiz at glencoe.com. Chapter 1 Get Ready for Chapter 1 3 Get Started on Chapter 1 You will learn several new concepts, skills, and vocabulary terms as you study Chapter 1. To get ready, identify important terms and organize your resources. You may wish to refer to Chapter 0 to review prerequisite skills. Study Organizer New Vocabulary English Español Tools of Geometry Make this Foldable to help you collinear • p. 5 • colineal organize your Chapter 1 notes about points, lines, and planes; angles and angle relationships; and formulas and coplanar • p. 5 • coplanar notes for distance, midpoint, perimeter, area, and congruent • p. 17 • congruente volume. Begin with a sheet of 11 × 17 paper. midpoint • p. 27 • punto medio 1 Fold the short sides to meet in segment bisector • p. 29 • bisectriz de segmento the middle. angle • p. 36 • angulo vertex • p. 36 • vertice angle bisector • p. 39 • bisectriz de un angulo perpendicular • p. 48 • perpendiculares 2 Fold the booklet in thirds polygon • p. 56 • poligono lengthwise. perimeter • p. 58 • perimetro volume • p. 69 • volumen 3 Open and cut the booklet in thirds lengthwise. Review Vocabulary ordered pair • p. P15 • par ordenado a set of numbers or coordinates used to locate any point on a coordinate plane, written in the form (x, y) 4 Label the tabs as shown. origin • p. P15 • origen the point where the two axes intersect at their zero points quadrants • p. P15 • cadrantes the four regions into which the x-axis and y-axis separate the coordinate plane x-coordinate • p. P15 • coordenada x the first number in an ordered pair y-coordinate • p. P15 • coordenada y the second number in an ordered pair y KY Math Online glencoe.com x-coordinate Study the chapter online coordenada x (x, y) y-coordinate Explore Math in Motion coordenada y Get extra help from your own Personal Tutor 0 x quadrant Use Extra Examples for additional help cadrant origin ordered pair Take a Self-Check Quiz origen par ordenado Review Vocabulary in fun ways Multilingual eGlossary glencoe.com 4 Chapter 1 Tools of Geometry Points, Lines, and Planes Why? On a subway map, the locations Then of stops are represented by points. You used basic geometric The route the train can take is concepts and properties to modeled by a series of connected solve problems. paths that look like lines. The flat surface of the map on Now which these points and lines lie Identify and model is representative of a plane. points, lines, and planes. Identify intersecting lines and planes. Points, Lines, and Planes Unlike the real-world objects that they model, shapes, points, lines, and planes do not have any actual size. In geometry, point, line, and plane are considered KY Program undefined terms because they are only explained using examples and descriptions. of Studies HS-G-S-SR6 Students will You are already familiar with the terms point, y analyze and apply spatial line, and plane from algebra. You graphed on relationships among points, a coordinate plane and found ordered pairs lines and planes (e.g., line collinear, coplanar). that represented points on lines. In geometry, plane these terms have a similar meaning. 0 x New Vocabulary undefined term The phrase exactly one in a statement such as, point point “There is exactly one line through any two line points,” means that there is one and only one. plane collinear coplanar For Your intersection Key Concept Undefined Terms definition defined term A point is a location. It has neither shape nor size. space Named by a capital letter " KY Math Online Example point A glencoe.com A line is made up of points and has no thickness or width. Extra Examples There is exactly one line through any two points. 1 Personal Tutor 2 Self-Check Quiz Named by the letters representing two points on the line m Homework Help or a lowercase script letter Example line m, line PQ or PQ , line QP or QP A plane is a flat surface made up of points that extends % infinitely in all directions. There is exactly one plane # through any three points not on the same line. Named by a capital script letter or by the letters naming $ three points that are not all on the same line K Example plane K, plane BCD, plane CDB, plane DCB, plane DBC, plane CBD, plane BDC Collinear points are points that lie on the same line. Noncollinear points do not lie on the same line. Coplanar points are points that lie in the same plane. Noncoplanar points do not lie in the same plane. Lesson 1-1 Points, Lines, and Planes 5 EXAMPLE 1 Name Lines and Planes Use the figure to name each of the following. a. a line containing point W 5 The line can be named as line n, or any two n ; of the four points on the line can be used to 7 name the line. 8 9 : VW WV VX XV VY YV P WX XW WY YW XY YX r b. a plane containing point X One plane that can be named is plane P. You can also use the letters of any three StudyTip noncollinear points to name this plane. plane XZY plane VZW plane VZX Additional Planes Although not drawn in plane VZY plane WZX plane WZY Example 1b, there is The letters of each of these names can be reordered to create other acceptable another plane that names for this plane. For example, XZY can also be written as XYZ, ZXY, ZYX, contains point X. Since points W, T, and X are YXZ, and YZX. In all, there are 36 different three-letter names for this plane. noncollinear, point X is also in plane WTX. ✓ Check Your Progress 1A. a plane containing points T and Z 1B. a line containing point T Personal Tutor glencoe.com EXAMPLE 2 Model Points, Lines, and Planes MESSAGE BOARD Name the geometric terms modeled by the objects in the picture. ( The push pin models point G. ) The maroon border on the card + models line GH. ' The edge of the card models line HJ. The card itself models plane FGJ. ✓ Check Your Progress Name the geometric term modeled by each object. 2A. stripes on a sweater 2B. the corner of a box Personal Tutor glencoe.com Intersections of Lines and Planes The intersection of two or more geometric figures is the set of points they have in common. Two lines intersect in a point. Lines can intersect planes, and planes can intersect each other.
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