""" >- .1iII ~- >- Each Chlpler cover&; a Clne tflll O1geometrtc m >- Exam,*,wl1ll UII1IIeIel, wtrlleel t_ UIIIt1n: , every ~ "' - mile ,olllem dril ProvenTecllliques Iroo1 ill Expert DAWN SUVA, Ph D How to Solve Word Problems in Geometry Dawn B Sova, Ph.D McGraw-Hili New York San Francisco Washington, D.C Auckland Bogota Caracas Lisbon London Madrid Mexico City Milan Montreal New Delhi San Juan Singapore Sydney Tokyo Toronto Library of Congress Cataloging-in-Publication Data Sova, Dawn B How to solve word problems in geometry / Dawn B Sova p cm Includes index ISBN 0-07-134652-X Word problems (Mathematics) Geometry-Study and teaching I Title QA461.S68 1999 99-37603 516 dc21 CIP McGraw-Hill A Division of The McGraw-HiUCompanies 'bZ Copyright © 2000 by The McGraw-Hill Companies, Inc All rights reserved Printed in the United States of America Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher DOC/DOC 9 ISBN 0-07-134652-X The sponsoring editor for this book was Barbara Gilson, the editing supervisor was Donna Muscatello, and the production supervisor was Tina Cameron It was set in Stone Serif by PRD Group Printed and bound by R R Donnelley &: Sons Company This publication is designed to provide accurate and authoritative information in regard to the subject matter covered It is sold with the understanding that neither the author nor the publisher is engaged in rendering legal, accounting, or other professional service If legal advice or other expert assistance is required, the services of a competent professional person should be sought -From a Declaration of Principles jointly adopted by a Committee of the American Bar Association and Committee of Publishers McGraw-Hill books are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs For more information, please write to the Director of Special Sales, McGraw-Hill, 11 West 19th Street, New York, NY 10011 Or contact your local bookstore This book is printed on recycled, acid-free paper containing a minimum of 50% recycled de-inked fiber Contents INTRODUCTION Chapter I-Points, Lines, Planes, and Angles Chapter 2-Deductive Reasoning 15 Chapter 3-Parallel Lines and Planes 27 Chapter 4-Congruent Triangles 41 Chapter 5-Quadrilaterals 57 Chapter 6-lnequalities in Geometry 69 Chapter 7-Similar Polygons 79 Chapter a-Right Triangles 91 Chapter 9-Circles 105 Chapter 10-Areas of Polygons and Circles 123 Chapter I I-Volumes of Solids 137 Chapter 12-Miscellaneous Problem Drill 145 Index 151 v Introduction Solving word problems in geometry is a challenge, but it can also be fun when you know how Not only you have to read and understand word problems and the basics of solving equations, but you also have to know the specialized vocabulary and symbols of geometry Some words, such as hypotenuse, isosceles, and secant, are unique to geometry, while others such as plane, line, and construction are everyday words that take on new meanings You also have to know and understand the symbols, and "translate" these symbols into words that allow you to write or think of geometry statements in everyday language Doing so will help you to "read" a diagram and to draw a diagram from given information without reading more into the diagram than actually exists Understanding and working geometry word problems takes practice The more types of word problems that you do, the better you will become in quickly deciding what the problem asks and reaching a solution In this book, you will find a step-by-step approach that gives you solutions to measure your increasing ability to work geometry word problems Once you have mastered the basics for solving geometry word problems, you will be able to easily apply these principles to even the most challenging advanced problems Chapter I POints, Lines, Planes, and Angles Points, lines, and planes are part of your everyday life When you watch a color television screen or the computer monitor, you are actually viewing a picture that is made up of hundreds of thousands of dots Because the dots are so small and so close together, your eyes see a complete picture, not the individual dots Each dot on the screen is similar to a point The point is the simplest figure that you will study in geometry Although it doesn't have any size, a point is usually represented by a dot that has size, and it is usually named by a capital letter All geometric figures consist of points One familiar geometric figure is a line, a series of connected points that extends in two directions without ending The picture of a line has some thickness, but the line itself has no thickness A line is referred to with a single lowercase letter, such as line g, if no points on the line are known If you know that a line contains points A and B, then you call it line AB or simply AB A plane is similar to a floor, wall, or tabletop Unlike these objects, however, a plane extends without ending and has no thickness Although a plane has no edges, it is usually pictured as a four-sided figure and labeled with a capital letter, plane R You can think of the ceiling and floor of a room as parts of horizontal planes, and the walls as parts of vertical planes Point, line, and plane are accepted as intuitive ideas in geometry that not require defining, but they are used in defining other terms in geometry Definitions to Know Acute angle An angle with measures between and 90° Adjacent angles (adj s) Two angles in a plane that have a common vertex and a common side but no common interior points Angle (4) The figure formed by two rays that have the same endpoint The rays are called the sides of the angle and their common endpoint is the vertex of the angle Bisector of an angle The ray that divides the angle into two congruent adjacent angles Bisector of a segment A line, segment, ray, or plane that intersects a line segment at its midpoint Collinear points Points all in one line Congruent Refers to objects that have the same size and shape Congruent angles Angles that have equal measures To indicate that two angles are congruent, write m4F = m4G or m4F == m4G (m4 = measure of angle) Congruent segments Line segments that have equal length To indicate that line segments AB and CD are congruent, write AB == CD or AB = CD Coplanar points Points all in one plane Intersection of two figures The set of points that are in both figures Length of a segment The distance between its endpoints, denoted as AB Length must be a positive number Midpoint of a segment The pOint that divides a line segment into two congruent segments Obtuse angle An angle with a measure between 90 and 180° Postulate In geometry, a statement that is accepted without proof Ray of a line Denoted AD, consists of a line segment AD and all other points P, such that D is between A and P The endpoint of AD is A, the point named first Right angle An angle with a measure of 90° Segment of a line Consists of two pOints on a line and all points that are between them It is denoted AD, in which A and D are the endpoints of the segment Space The set of all points Straight angle An angle with a measure of 180° Theorem In geometry, an important statement that can be proved Relevant Postulates and Theorems Postulate I (Ruler Postulate) The points on a line can be paired with the real numbers in such a way that any two points can have coordinates and l Once a coordinate system has been chosen in this way, the distance between any two points equals the absolute value of the difference of their coordinates Postulate (Segment Addition Postulate) If B is between A and C, then AB + BC = AC Postulate J (Protractor Postulate) On AB in a given plane, choose any point between A and B Consider OA and OB and all the rays that can be drawn from on one side of AB These rays can be paired with the real numbers from to 180 in such a way that (a) (b) OA is paired with 0, and OB with 180 If OP is paired with X, and OQ with y, then m4POQ = Ix - yl Postulate (Angle Addition Postulate) If point B lies in the interior of 4AOC, then m4AOB + m4BOC = m4AOC If 4AOC is a straight angle and B is any point not on AC I then m4AOB + m4BOC = 180° Postulate A line contains at least two points; a plane contains at least three points not all in one line; space contains at least four points not all in one plane Postulate Through any two pOints there is exactly one line 41 What is the area of a right trapezoid with bases of lengths 16 and 22, and sides of lengths and 10? How many squares with sides of length can you fit into the area of a rectangle with sides of lengths 18 and 27? If a book distributor wants to save on the cost of shipping multiple copies of the same book in cardboard cartons, what is the minimum volume of a carton needed to hold 40 copies of a book having a width of inches, length of inches, and thickness of inches? What is the side measure of a hexagon inscribed in a circle having a diameter of 14? How many rings measuring feet in radius can a circus fit onto a floor measuring 36 feet wide and 72 feet long? If the sum of two angles of a rhombus is 190°, then the sum of the remaining two angles is - What volume of water must be used to fill a cylindrical pool with a height of feet and a diameter of 40 feet? A skater wants to find how many more feet she is skating each day if she skates seven times around the perimeter of a circular rink than if she simply skates 10 round trips on the diameter of the rink, which is 40 feet Use 22/7 to approximate the value of 1T If the perimeters of two circles are in a ratio of 4:1, then their radii are in a ratio of If the volume of a cube is 1331 cm , what is the area of a side? 42 43 44 45 46 47 48 49 50 I 10 II 12 ANSWERS 58° 45°,45° 30° 20 2r Ilr 15.5° 90° Noon or midnight (both hands are on the 12) 60° 50 feet 43 148 13 72 meters 14 150 feet 15 52 feet 16 10017 17 12 by 16 18 80 yards 19 312 tiles 20 242 yards 21 15 22 70°,70° 23 15 24 28v15 25 32 26 34317 27 63° 28 Ilr,68° 29 817 30 280 31 441 m3 32 12817 m3 33 67° 34 35 feet 35 15 feet 36 48° 37 285° 38 60° 39 40 128V3 41 152 42 54 43 4320 in 44 45 18 46 170° 47 320017 ft3 48 80 feet 49 4:1 50 121 cm 149 Index Addition Property, 18 Algebra properties, 18-20 Angle Addition Postulate,S Angle-Angle-Side Postulate, 48 Angle-Side-Angle Postulate, 47 Angles, 3-13 acute, 4, 21 adjacent, 4, 20, 21 bisectors of, 4, 20, 49, 83 complementary, 4, 21 congruent, 4, 21 corresponding, 27 obtuse, right, 4, 15 supplementary, 18,21 Apothems of regular polygons, 123 Arcs: addition of, 108 major, 105, 107, 110 minor, 105, 107, 109, 110 Bisectors: perpendicular, 46, 48-49, 128 Central angles, 105, 124 measure of, 124 Chords, 105, 109, 110 intersecting, 110 Circles, 105-121 areas of, 126, 127, 131, 132 Circles (cont.): circumscribed, 106 concentric, 107 diameter, 107, 109 radius of, 107, 108 semicircles, 108 tangent, 108 Cones, 139 volume of, 140, 142 Congruency, 124 Cylinders, 139 altitudes of, 138 volume of, 140, 141 Deductive reasoning, 15-26 conclusion, 16 conditional statements, 16, 17 converse, 16 definition, 16 hypothesis, 16 if-then statement, 16 proofs, 17-18 Division Property, 19 Exterior angles: alternate, 32, 33, 35 of a triangle, 69 Formal proofs, 18-23 Hypotenuse-Leg Postulate, 48 151 Indirect proof, 70 Inductive reasoning: counterexample, 15 Inequalities, 69-78 properties, 70-71 theorems, 71 Inscribed angles, 107 measure of, 110 Inscribed polygons, 110 Interior angles: alternate, 27, 28, 31 remote, 70 Lines, 3-13, 27 parallel, 28, 29, 30, 59, 83 perpendicular, 17, 20, 31, 46, 108 skew, 28 Means, geometric, 91, 95-96, 98 Medians: of trapezoids, 58_ of triangles, 46 MidpOints: of segments, 20, 60 of triangles and trapezoids, 59 Multiplication Property, 19 Parallelograms, 57, 59, 84 areas of, 125 bases of, 124 diagonals of, 58, 59 Planes, 3-13 intersecting, parallel, 28, 30 Points, 3-13 collinear, coplanar, 4, Polygons, 79-89 area of, 123-131 height of, 124 inscribed, 107 perimeter of, 124 radius of, 124 regular, 124 Prisms, 139 altitude of, 137 bases of, 138 lateral faces of, 139 types of, 138 volume of, 140 Proportion, 80 extended, 80 extremes of, 80, 98 means of, 80, 96 means-extremes of, 80, 85-86 Pyramids: altitudes of, 138 152 Pyramids (cont.): regular, 139 slant height of, 139 volume, 140, 141 Pythagorean Theorem, 91-103 Quadrilaterals, 57-67 angles of, 59 inscribed, 109 Radicals, 91,94-95 Ratio, 81 Rays, 4, Reasoning: deductive, 15 inductive, 15 Rectangles, 58 areas of, 125, 127 volume of, 140 Reflexive Property, 19 Rhombuses, 58 areas of, 125, 128 diagonals of, 129 Right triangles, 42-43, 48, 91-103 altitudes of, 92 formulas for, 92 hypotenuse of, 48, 60, 91, 92, 97-99 sides, 92 special, 93 theorems, 92-93 Scale, 81 factor, 81 Secants, 108 intersecting, 110 and tangents intersecting, 111 Segments, bisectors of, 4, 46 congruent, intersecting, of a line, secant, 111 secant and tangent, 111 Side-Angle-Side Postulate, 47 Side-Side-Side Postulate, 47 Sides: corresponding, 44 of triangles, 46, 47, 59, 82, 91, 92 Solids, volumes of, 137-144 Spheres, 139 volume, 140, 142 Squares, 58 areas of, 124 perimeters of, 126 Statements: contrapositives of, 69 converses of, 16 if-then, 16 Statements (cant.): inverses, 70 logically equivalent, 70 Subtraction Property, 19 Substitution Property, 19 Symmetric Property, 20 Tangent: common, 106 congruent, 109 intersecting a circle, 107, 108 Transitive Property, 20 Transversals, 29, 31, 32 Trapezoids, 57, 58 areas of, 126, 129-130 isosceles, 58 Triangles: acute, 42, 92 Triangles (cont.): altitude, 44, 91 areas of, 125 classifying, 41 congruent, 48, 82 equiangular, 42-43 equilateral, 41-42 isosceles, 41-42, 47 obtuse, 42-43, 92 right (see Right triangles) scalene, 41-42 sides of (see Sides, of triangles) Venn diagram, 70 Vertex (see Vertices) Vertical angles, 18, 20 Vertices, 41 153 ... Juan Singapore Sydney Tokyo Toronto Library of Congress Cataloging -in- Publication Data Sova, Dawn B How to solve word problems in geometry / Dawn B Sova p cm Includes index ISBN 0-07-134652-X Word. .. the angle into two congruent adjacent angles Bisector of a segment A line, segment, ray, or plane that intersects a line segment at its midpoint Collinear points Points all in one line Congruent... is a line Theorem I If two lines intersect, then they intersect in exactly one point Theorem Through a line and a point not in the line there is exactly one plane Theorem If two lines intersect,