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McGraw-Hill’s 10 ACT Practice Tests ABOUT THE AUTHOR Steve Dulan has been involved with the ACT since 1982, when he received a score of 32 on his own test as a high school junior at Iron Mountain High School That score qualiﬁed him for the State of Michigan Competitive Scholarship in 1983 In 1989, after serving as a U.S Army infantry Sergeant, and during his time as an undergraduate at Michigan State University, Steve became an ACT instructor He has been helping students to prepare for success on the ACT and other standardized exams ever since Steve attended The Thomas M Cooley Law School on a full Honors Scholarship after achieving a 99th percentiIe score on his Law School Admission Test (LSAT) In fact, Steve scored in the 99th percentile on every standardized test he has ever taken While attending law school, Steve continued to teach standardized test prep classes (including ACT, SAT, PSAT, GRE, GMAT, and LSAT) an average of thirty hours each week, and tutored some of his fellow law students in a variety of subjects and in essay exam writing techniques Steve has also served as an instructor at the college and law school levels Thousands of students have beneﬁted from Steve’s instruction, coaching, and admissions consulting and have gone on to the colleges of their choice His students have gained admission to some of the most prestigious institutions in the world and received many scholarships of their own A few of them even beat his ACT score! Since 1997, Steve has served as the President of Advantage Education (www.AdvantagEd.com), a company dedicated to providing effective and affordable test prep education in a variety of settings, including classes and seminars at high schools and colleges around the country, summer College Prep Camps at The University of Michigan, and one-on-one via the Internet worldwide Copyright © 2008 by The McGraw-Hill Companies, Inc Click here for terms of use McGraw-Hill’s 10 ACT Practice Tests SECOND EDITION Steven W Dulan and the faculty of Advantage Education New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto Copyright © 2008 by The McGraw-Hill Companies, Inc All rights reserved Manufactured 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 0-07-159147-8 The material in this eBook also appears in the print version of this title: 0-07-159146-X All trademarks are trademarks of their respective owners Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark Where such designations appear in this book, they have been printed with initial caps McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs For more information, please contact George Hoare, Special Sales, at george_hoare@mcgraw-hill.com or (212) 904-4069 TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc (“McGraw-Hill”) and its licensors reserve all rights in and to the work Use of this work is subject to these terms Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill’s prior consent You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited Your right to use the work may be terminated if you fail to comply with these terms THE WORK IS PROVIDED “AS IS.” McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE McGraw-Hill and its licensors not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom McGraw-Hill has no responsibility for the content of any information accessed through the work Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise DOI: 10.1036/007159146X Professional Want to learn more? We hope you enjoy this McGraw-Hill eBook! If you’d like more information about this book, its author, or related books and websites, please click here For more information about this title, click here CONTENTS Acknowledgments vii Introduction: Using This Book How to Use the Practice Tests A Note on Scoring the Practice Tests ix ix ix UNDERSTANDING THE ACT What Is the ACT? Who Writes the ACT? Registering for the ACT Why Do ACT Exams Exist? ACT Scores Writing Test Scoring Guidelines PRACTICE TEST Answer Key Scoring Guide Answers and Explanations PRACTICE TEST Answer Key Scoring Guide Answers and Explanations PRACTICE TEST Answer Key Scoring Guide Answers and Explanations PRACTICE TEST Answer Key Scoring Guide Answers and Explanations PRACTICE TEST Answer Key Scoring Guide Answers and Explanations 1 3 61 65 69 91 145 149 153 175 230 233 237 257 309 313 317 339 394 397 401 vi CONTENTS PRACTICE TEST Answer Key Scoring Guide Answers and Explanations PRACTICE TEST Answer Key Scoring Guide Answers and Explanations PRACTICE TEST Answer Key Scoring Guide Answers and Explanations PRACTICE TEST Answer Key Scoring Guide Answers and Explanations PRACTICE TEST 10 Answer Key Scoring Guide Answers and Explanations 421 477 481 485 509 563 567 571 593 651 655 659 683 739 743 747 769 822 825 829 ACKNOWLEDGMENTS The author would like to acknowledge the contribution of the faculty and staff of Advantage Education You are not only the smartest, but also the best Special thanks to Lisa DiLiberti, Amy Dulan, Matt Mathison, Kathy Matteo, Blair Morley, Ryan Particka, Andrew Sanford, Kim So, and Amanda Thompson All of you put in extra effort to make this book a success Copyright © 2008 by The McGraw-Hill Companies, Inc Click here for terms of use This page intentionally left blank 834 PRACTICE TEST 10 ANSWERS AND EXPLANATIONS a run-on sentence Both of these choices should be eliminated Only answer choice H clearly indicates that weaponry, ammunition, and materials have all changed through the centuries see if it makes sense and effectively conveys the idea Answer choices F, G, and H are very wordy and awkward, and they not express the idea as clearly and simply as does answer choice J 65 The best answer is A This question tests your ability to express the idea clearly and simply Since “silk” is the material being discussed, it makes sense to place the underlined portion directly after the word “silk,” as it is written Placing the underlined portion elsewhere makes the sentence awkward and unclear 71 The best answer is B This question requires you to correctly punctuate the underlined portion The sentence as it is written is a run-on sentence, so eliminate answer choice A Answer choice C creates an incomplete sentence, so it should also be eliminated A semicolon should be followed by an independent clause, so eliminate answer choice D 66 The best answer is H This question requires you to select the best transition between sentences The word “soon” implies a sense of time that does not ﬁt within the context of the paragraph Eliminate answer choice F The paragraph discusses the importance of protecting the wearer from bullets It makes sense that, because the ﬂak-jacket did not protect the wearer from bullets, that the best transition word to use is “unfortunately.” 72 The best answer is J This question tests your ability to express the idea clearly and simply, which, in this case, means eliminating answer choices that include irrelevant information Since the sentence already indicates that the impact is spread around the body, any information about how this is in contrast to the impact being sustained in one area is redundant and irrelevant Omitting, or removing, the underlined portion keeps the sentence simple, while still effectively expressing the idea 67 The best answer is D This question tests your ability to recognize redundancy The word “today” suggests “recent” and “modern,” so it is not necessary to include all of those words in the sentence The most simple way to express the idea is with the word “today’s.” 68 The best answer is G The best way to answer this question correctly is to insert the answer choices in the sentence and select the one that makes sense You must answer the question, “A variety of synthetic materials, some of which are” what? The word “effective” best answers that question Answer choices F, H, and J are very awkward and not successfully convey the idea being presented in the sentence 69 The best answer is C The preceding sentence discusses the “production” of synthetic materials, so you should look for an answer choice that further explains the production of these materials Eliminate answer choices A and D because they not discuss production of material You must also tie the idea presented in the preceding sentence to the concept of cost, since that is mentioned in the sentence about which you are being asked Answer choice C does this best 70 The best answer is J This question tests your ability to express the idea clearly and simply In this case, you should try the shortest answer choice to 73 The best answer is B The question asks you to indicate that body armor does not always prevent injury or death Answer choices A, C, and D suggest that the material in a bullet-proof vest will often or always prevent injury or death Answer choice B effectively shows that, only under some circumstances, but certainly not always, will body armor prevent injury or death 74 The best answer is F The best way to answer this question correctly, is to determine what, if anything, the underlined portion adds to the sentence The underlined portion includes information that indicates that the bullet-proof-vest wearer could sustain serious injury while wearing the vest If this information is removed, the paragraph would lose a statement that suggests protective gear does not guarantee or ensure a lack of serious injury The other answer choices are not supported by the context of the paragraph 75 The best answer is C This question requires you to select the correct verb form The verb “entrust” means to “place trust in,” which makes sense based on the context of the paragraph The idea being conveyed is that police ofﬁcers and military personal know that their safety cannot be “entrusted” completely to the protective gear that they wear; they must take additional precautions to ensure their safety PRACTICE TEST 10 ANSWERS AND EXPLANATIONS Mathematics Test Explanations The correct answer is D Given that angle A is 20◦ and that angle B is three times larger than angle C, you can set up the following equation 20◦ + B + C = 180◦ Since B = 3C, you have 20◦ +3C+C = 180◦ Combine like terms: 20◦ +4C = 180◦ Solve for C: 4C = 160◦ ; C = 40◦ If C = 40◦ , then B = (40◦ )(3) = 120◦ The correct answer is G The ﬁrst step is to calculate the total cost of the trip by multiplying the number of people in the band (65) by the cost per member: 65 × 18.50 = $1202.50 A total of $585.00 was raised from the bake sale, $1202.50 − $585.00 = $617.50, which is the total amount left to earn That means that each of 617.50 the 65 members must pay an additional , 65 or $9.50 The correct answer is B To solve this problem, set up an equation showing that Fred’s total daily income equals his ﬁxed daily income ($40.00) + the amount he earns for washing cars Set the number of cars equal to C: $61.00 = $40.00 + $1.75C Solve for C: $1.75C = $21.00 C = 12 The correct answer is G Since you are given the s equation for molality, m = , and you are given k m = 0.2 and k = 13, plug in the values and solve: s 0.2 = 13 s = 2.6 The correct answer is C The best approach to this problem is to identify the relationships between the angles Since a and b are vertical angles formed from the intersection of parallel lines and transversals, they are equal in measure Since the same transversal (l1 ) creates angle x, angle x is equal to angles a and b Eliminate answer choices A and B Angle c is a supplementary to angle x, which means that angle c + angle x = 180◦ There are no right angles in the ﬁgure, so angle c is NOT equal to angle x; answer choice C is correct The correct answer is F The expression 4.2×10−5 is in scientiﬁc notation The −5 exponent means that the decimal point has been moved ﬁve spaces 835 to the right of its original position So, move the decimal ﬁve places to the LEFT back to its original position: 0.000042 The correct answer is E Multiplying by 0.01 is the same as dividing by 100, since 0.01 = 100 Move the decimal place two spaces to the left to get 0.03234 The correct answer is K To answer this question, you need to ﬁrst expand the equation and then simplify it: x − 2x + x−1 (x − 1)(x − 1) x−1 The (x − 1) from the numerator cancels out the (x − 1) in the denominator, so the equation is equal to x − The correct answer is A This simple algebra problem asks you to solve for x Combine like terms and isolate x on one side of the equation − x + = 19 −x + 11 = 19 −x = x = −8 x = −2 10 The correct answer is J With the information given, you can ﬁnd that the measure of angle WZY = 180◦ − (25◦ + 55◦ ) = 100◦ Since WXY is opposite WZY in the same parallelogram, it also equals 100◦ 11 The correct answer is D Since the circle at −1 is closed, that means that −1 should be included Eliminate answer choices A, B, and E, which not include –1 Since the graph goes to the right, the inequality must be greater than or equal to −1 12 The correct answer is G Solve for x, and keep track of the negative signs 93 − x = 342 −x = 342 − 93 −x = 249 x = −249 13 The correct answer is E You are given that AB = and BX = 10 Use the Pythagorean Theorem to solve for AX: 62 + (AX)2 = 102 (AX)2 = 64 AX = 836 PRACTICE TEST 10 ANSWERS AND EXPLANATIONS If AX = and XC = 4, then AC must equal 12 Given that AB = and AC = 12, again use the Pythagorean Theorem to solve for BC: 62 + 122 = (BC)2 180 = (BC)2 √ BC = 180 = √ (36)(5) = 14 The correct answer is F The formula for the area of a rectangle is A = l × w, with l as the length and w as the width The question states that the length of the parking lot is three times as long as the width If w is the width, then the length is 3×w, or 3w Therefore, the area of the parking lot is 3w × w = 315 15 The correct answer is C To solve this problem, set up a proportion: 3.5 miles is to x miles as 40 minutes is to 60 minutes Cross-multiply and solve for x: 3.5 40 = x 60 40x = 210 x = 5.25 16 The correct answer is K First, divide the region into two rectangles, as shown: 19 The correct answer is B The ﬁrst step in answering this question is to rearrange one of the given equations so that it is expressed in terms of y: y + 3x = y = −3x + To solve for x, the next step is to plug −3x + in for y in the second equation: 2x − (9 − 3x) = 2x − + x = 3x = + x = 20 The correct answer is K When multiplying fractions, multiply the numerators together and then multiply the denominators together and then reduce 20 = = 12 2 = 3 When dividing fractions, multiply the ﬁrst fraction by the reciprocal of the second fraction and reduce ÷ The ﬁrst rectangle is feet by 16 feet The second rectangle is 10 feet by 14 feet (19 feet minus feet) The area of a rectangle is calculated by multiplying the length by the width Add the areas together: Rectangle 1: (5)(16) + Rectangle 2: (10 (14) = 220 square feet 17 The correct answer is A Given 32 ounces in a quart, you can calculate that there are 64 ounces in quarts quarts, or 64 ounces, costs $2.65 Divide $2.65 by 64 (2.65 ÷ 64) to get $0.04 per ounce 18 The correct answer is H Angles x and y are supplementary angles, so their sum is 180◦ Angle x is given as 125◦ , so angle y is 180◦ − 125◦ = 55◦ 15 45 = 21 The correct answer is C To solve this problem, combine like terms Remember to keep track of the negative signs! x − (3x − 2) + 2x(4x − 1) x − 3x + + 8x − 2x 9x − 5x + 22 The correct answer is G First, factor the numerator and denominator: x − 6x + (x − 3)(x − 3) = 6(x − 3) 6x − 18 Next, cancel out an (x − 3) from the numerator and (x − 3) denominator, which leaves you with 23 The correct answer is C You are given that x = is one solution, which means that (x − 3) is one of the factors of 2x −5x−a = Since the coefﬁcient of the x –term is 2, you can set up the other factors PRACTICE TEST 10 ANSWERS AND EXPLANATIONS as follows: (2x + )(x − 3) The x terms must equal –5, so apply the FOIL method to ﬁnd the missing term: −6x + x = −5x −6x + x = −5x; the missing term must be Use FOIL again to solve for a: (2x + 1)(x − 3) 2x − 6x + x − 2x − 5x − 28 The correct answer is F The easiest way to solve this problem is to cross-multiply and solve for x: 4x 4x = x−1 2x + 4x(2x + 2) = 4x(x − 1) 8x + 8x = 4x − 4x; combine like terms 4x + 12x = 0; simplify by factoring out 4x 4x(x + 3) = 4x = 0, so x = x + = 0, so x = −3 The two solutions are and –3 The sums of these solutions is + −3, or −3 Thus, a = 24 The correct answer is F You are given that angle P is 55◦ Since PQR is a right triangle, angle Q is 90◦ So, angle R must be 180◦ − (55◦ + 90◦ ) or 35◦ 25 The correct answer is E You are given that gallon of paint can cover 36 square feet The room is feet by 12 feet, or 96 square feet Set up a proportion: x = 36 96 Cross-multiply and solve for x: 36x = 96 x = approximately 2.666 The question asks you to round to the nearest whole gallon, so round up to gallons 26 The correct answer is G You are given that x ≤ y and y ≤ z If x = y and y = z, then x = z could be true Eliminate answer choices F, J, and K, which show that Roman Numeral I (x = z) is NOT true If x < y and y < z, then x < z could be true But x > z could not be true in either case So only II CANNOT be true, answer choice G 27 The correct answer is B The sine of an angle is the length of the opposite side divided by the opp length of the hypotenuse You are given the hyp measure of the opposite side, but you are not given the hypotenuse So you must use the Pythagorean Theorem to ﬁnd the measure of the hypotenuse: (1)2 + (5)2 = (hypotenuse)2 + 25 = (hypotenuse)2 √ 26 = (hypotenuse)2, so the hypotenuse = 26 Now, you can ﬁnd sin C: 837 opp hyp =√ 26 29 The correct answer is B This problem requires you to understand absolute value Absolute value is the distance of the given value from zero on the number line, without regard to its sign So, |−2|2 + | − 5| − = 22 + − 3, which is + − = 30 The correct answer is G Lines XY and ZW are parallel and are bisected by line AB As you can see in the diagram from the question, these lines form two triangles Because the lines XY, YZ, and ZW are equal, the sides of the triangles that are created are also equal The triangle with ∠s is a reﬂection of the triangle with ∠t, so ∠s = ∠t If ∠t = 45◦ , then ∠s = 45◦ 31 The correct answer is C Here, you must ﬁrst solve the inequality: x − ≤ 13 x ≤ 16 Because a negative number squared will yield a positive result, you must consider both the positive and negative values of x x ≤ and x ≥ −4 The greatest real value of x is if x = 32 The correct answer is F By deﬁnition, an isosceles triangle has a hypotenuse, c, and two legs, a and b, that are congruent Use the measure of the hypotenuse given in the problem (12), and the Pythagorean theorem, a2 + b2 = c2 , to determine the length of one of the legs Remember that a = b: a2 + a2 = 122 2a2 = 144 a2 =√72 a = 72 √ √ The 72 can be reduced to 838 PRACTICE TEST 10 ANSWERS AND EXPLANATIONS 33 The correct answer is C The formula for a circle is (x − a)2 + (y − b)2 = r , where (a, b) is the center of the circle and r is the radius of the circle The question gives you the equation of the circle as (x − 2)2 + (y + 1)2 = So, a is and b is −1 The center of the circle, (a, b), is (2, −1) 34 The correct answer is J Find the slope by putting 2x − 3y = into the slope-intercept form (y = mx + b where m is the slope): −3y = −2x + 2 y = x − The slope is 3 35 The correct answer is D This question is asking you to determine a weighted average To answer the question, ﬁrst calculate the total number of TVs: (2 × 5), (3 × 20), (4 × 15), and (5 × 10) Next, add these values together and divide by the total number of houses (50): (2 × 5) + (3 × 20) + (4 × 15) + (5 × 10) 50 180 10 + 60 + 60 + 50 = = 3.6 50 50 36 The correct answer is H Since you are given the radii of both circles, you can ﬁnd the area of both circles The radius of the smaller circle is 2, so the area is π(2)2 , or 4π The radius of the larger circle is 6, so the area is π(6)2 = 36π Since the smaller circle is removed from the larger circle, simply subtract the area of the smaller circle from the area of the larger circle: 36π − 4π = 32π 37 The correct answer is B By deﬁnition, the slope change in y The question gives the change in x slope of the line as , which means that for every positive change of in y, there is a positive change in x of The question asks you to determine the x-intercept, or where y = You are given a point on the line, (−2, 2) Use the point-slope form of a line, y = mx + b, set y to 0, m to , and solve for x: (0 − 2) = (x + 2) −2 = x + 3 of any line is = x+ 3 10 − = x, x = −5 The x-intercept is (−5,0) 3 − 38 The correct answer is J From the graph, you can determine that the inequality is x ≥ −2 However, that is not an answer given So, you must solve the inequalities in the answer choices Answer choices F and K not contain a greater than or equal to sign, so they may be eliminated Solve the remaining three inequalities: ⎧ 4x − ≥ 2x − ⎪ ⎪ ⎪ ⎪ ⎪ ⎨2x ≥ −1 Answer choice G ⎪ x≥− ⎪ ⎪ ⎪ ⎪ ⎩ Eliminate answer choice G ⎧ 5x + ≥ x ⎪ ⎪ ⎨ Answer choice H 4x ≥ −5x ≥ − ⎪ ⎪ ⎩ Eliminate answer choice H ⎧ ⎪ ⎨3x − ≤ 5x + Answer choice J −2x ≤ ⎪ ⎩ x ≥ −2 (remember to switch the sign when dividing by a negative number.) Answer choice J is correct 39 The correct answer is D By deﬁnition, the slope change in y of any line is The question asks you to change in x determine the slope from two points on the graph, so divide the change in y between the two points given by the change in x between the two points given: y1 − y2 5−3 = = x1 − x2 5−1 = 40 The correct answer is K You are given two of the three unknowns; plug I = 2.25 and r = 0.03 (the decimal equivalent of 3%) into the formula I = Pr: 2.25 = P(0.03) Solve for P: P = 75 PRACTICE TEST 10 ANSWERS AND EXPLANATIONS 41 The correct answer is E Graph the rectangle on a coordinate graph 839 46 The correct answer is J To solve this problem, replace y with y − 2, and z with z + 1: x = 3(y − 2) − 4(z + 1) + = 3y − − 4z − + = 3y − 4z − So, you went from x = 3y−4z+7 to x = 3y−4z−3 Subtract the two to see the difference: (3y − 4z + 7) − (3y − 4z − 3) 3y − 3y = −4z − (−4z) = − (−3) = 10 As you can see from the graph, the fourth vertex must be (−2, 2) 42 The correct answer is F Perpendicular lines have slopes that are the negative reciprocal of each other Therefore, if one of the lines has a slope of − , 7 the slope of the other line must be 43 The correct answer is D You are given that number of jars of jelly, j, is three less than twice the number of jars of applesauce, a Put this into equation form: j = 2a − Since none of the answers match this one, rearrange the equation: j + = 2a 44 The correct answer is G The ﬁrst step in answering this question is to factor the equation 3x − 5x + = 0: (3x − 2)(x − 1) = To ﬁnd the two solutions to this equation, solve for zero: 3x − = 3x = 2 x = (solution 1) x−1=0 x = (solution 2) The solutions to the equation given are x = and x= 45 The correct answer is A To solve this problem, you must ﬁrst recall that the maximum value of the sine function is 1, and that the smallest value for a π π at this maximum value will be So, if 2a = , 2 π then a = Since the difference is a positive 10, that means that the original value of x was 10 greater than the new value of x Thus, the value of x decreased by 10 47 The correct answer is B The ﬁrst step in answering this question is to determine what we will call your critical numbers Solve for x as if there were an equal sign instead of an inequality sign: (x − 1)(4 − x) = x − = and − x = x = and x = Only answer choice B has both of these numbers, so answer choice B is correct To make sure, choose a number that is greater than 4, like 5, and see if it is a solution to the inequality (x − 1)(4 − x) < 0: (4)(−1) = −4 < 0; x > is correct Choose a number that is less than 1, like 0: (−1)(4) = −4 < 0; x < is also correct 48 The correct answer is H You are given that the triangle is an isosceles right triangle, which means that it is a 45◦ −45◦ −90◦ triangle One of the characteristics of this √ type of triangle is that the hypotenuse is times the measure of each of the √ legs So the ratio of the hypotenuse to PQ is : 49 The correct answer is D The tangent of any acute angle is the opposite side divided by the length of opp Cosine is equal to the the adjacent side adj length of adjacent side divided by the hypotenuse opp adj Since tan x = , then the adjahyp adj cent side must be 4; eliminate answer choices B, C, and D Now, ﬁgure out the measure of the hypotenuse by using the pythagorean theorem: 32 + 42 = (hyp)2 840 PRACTICE TEST 10 ANSWERS AND EXPLANATIONS + 16 = (hyp)2 25 = (hyp)2 = the hypotenuse So, cos x = 50 The correct answer is J This question requires you to factor out the (a+b) so that you end up with an equivalent, but simpler, equation The equaa+b can be simpliﬁed in the tion b(a + b) − 2a(a + b) following way: a+b 1 × = a + b b − 2a b − 2a By factoring out the (a+b), the equation is reduced to b − 2a 51 The correct answer is D You know that QRX is a right triangle, and QR = Since X is the midpoint√ of SR, which is 8, RX must equal QX = 2, since √ RX is an isosceles triangle where of the legs the hypotenuse is times the measure √ 2, PX must also of the triangle Since QX = √ the perimeter of triangle PXQ is equal √ So, √ √ + + 2, or + 52 The correct answer is K The question states that the slope of the line is By deﬁnition, the slope of y1 − y2 change in y , so set equal to a line is the change in x x1 − x2 and solve for t: − (−2) = 3−t = 3−t (3 − t) × = 2(3 − t) = 18 3−t =9 −t = 6, or t = −6 53 The correct answer is A To solve this problem, actual base model base set up a proportion = actual height model height 32 = 56 x Cross-multiply and solve for x: 32x = 448 x = 14 54 The correct answer is H The ﬁrst step in answering this question is to rearrange the terms so that the x’s are all on one side of the inequality, −5x − > 3x + 1: −7 > 8x + The next step is to get x alone: −8 > 8x −1 > x, which is equivalent to x < −1 55 The correct answer is C The ﬁrst step in answering this question is to use the Pythagorean Theorem, a2 + b2 = c2 , to determine the length of AB BD, the line that bisects AC, creates a second triangle, BAD Use the length of lines BD and AD to determine the length of AD (let AD = a) a2 + 42 = 52 a2 + 16 = 25 a2 = a=3 Because you known that BD bisects AC, you know that the length of DC must also equal 56 The correct answer is G First, solve the equation for x: 2x + x−2= 3x − = 2x + x = 11 Only the interval 11 ≤ x
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