Sharon Serrago, Inc.

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1 Sharon Serrago, Inc. SAT/PSAT Study Materials Preparing students for higher SAT/PSAT scores Co-authors Elizabeth Howell Shelly Vorwerk Ross Evans 850 N. Dorothy Dr., Suite 50 Richardson, TX fa SAT and PSAT are registered trademarks of the College Entrance Eamination Board. The College Entrance Eamination Board is not involved in the production of these materials.

2 TABLE OF CONTENTS I. Test Taking Tips and Gridding... 9 II. Heart of Algebra A. Substitution and simplification of algebraic epressions.. 19 B. Linear equations and inequalities Solving linear equations.. 3. Writing linear epressions and equations Solving linear inequalities C. Function notation D. Absolute value E. Lines in the coordinate plane F. Systems of linear equations and inequalities Systems of linear equations. 7. Systems of linear inequalities.. 8 III. Passport to Advanced Math A. Eponents and radicals.. 87 B. Polynomials C. Rational epressions and equations Rational epressions Rational equations D. Radical equations E. Comple numbers.. 11 F. Quadratic functions and equations Solving by factoring Equivalent forms of quadratic epressions Connecting quadratic functions to graphs Solving by the quadratic formula G. Graphs of polynomials H. Eponential functions and equations I. Comparing linear and eponential growth J. Modeling with nonlinear equations K. Systems of nonlinear equations L. Literal equations IV. Problem Solving and Data Analysis A. Ratio, proportion, and percent Ratio and proportion Copyright 017 by Sharon Serrago, Inc. 3

3 . Direct and inverse proportions Percentages B. Unit conversion C. Data interpretation D. Measures of center E. Measures of spread... 4 F. Sampling and inference Sample size Inference and sample bias G. Scatterplots H. Probability Simple probability Conditional probability V. Additional Topics in Math A. Geometric notation B. Lengths and midpoints C. Angles D. Triangles E. Congruence and similarity F. Quadrilaterals G. Area and perimeter H. Regular polygons I. Circles Fundamentals of circles Equations of circles J. Solid geometry K. Trigonometry Trigonometric ratios Additional trigonometric topics Radian measure VI. Number and Operations Review A. Properties of integers B. Arithmetic word problems C. Squares and square roots D. Fractions and rational numbers E. Elementary number theory factors, multiples, remainders, prime numbers. 375 F. Sequences G. Sets union, intersection, elements H. Counting problems I. Logical reasoning Copyright 017 by Sharon Serrago, Inc. 4

4 VII. Writing A. Subject-verb agreement B. Noun-noun agreement C. Pronoun usage D. Introductory prepositional phrases. 438 E. Parallelism F. Fragments, comma-splices, and run-ons G. Semicolons H. Colons I. Dashes 471 J. Participial phrases K. Apostrophes L. Coordination and subordination M. Appositives N. Adjectives and adjective clauses O. Adverbs and adverb clauses P. Active vs. passive voice Q. Tense. 545 R. Comparisons S. Shifts in mood T. Noun clauses. 57 U. Commas with dates, addresses, and series V. Frequently confused words VIII. Reading A. Types of reading questions 1. Emphasis on words in contet Emphasis on command of evidence Inclusion of informational graphics Specified range of tet compleity B. Specific reading requirements C. Tips and strategies for reading D. Practice reading passages 1. Great Epectations Silas Marner # Silas Marner # Learning about Our Solar System The Minister s Black Veil Deforestation via Beetles Global Warming Oliver Twist Benjamin Franklin s The Way to Wealth Part I Aleander Hamilton s Federalist Papers No Thomas Paine s Common Sense Andrew Jackson s First Inaugural Address James Madison s First Inaugural Address. 655 E. Answers to reading passages.. 66 Copyright 017 by Sharon Serrago, Inc. 5

5 Rational Epressions and Equations A rational number is any number that can be written as the ratio of two integers. Eamples would include 1, 47, 1.3, or A number doesn t actually have to be written in fraction form to be rational. All integers are rational numbers. A rational epression is a ratio of two polynomials. Eamples include 3 5 4,, or 3. 7 Rational Epressions To simplify a rational epression, common factors in the numerator and denominator can be cancelled. 4 0 Eample 1 Simplify C) 5 B) D) Eplanation 4 0 First, factor both the numerator and denominator: 5 4 Cancelling 5 leaves, answer choice B. 5 becomes 4( 5). ( 5)( 5) To multiply rational epressions, cancel first, then multiply. Eample Simplify 56 y y 4 1 4y C) y B) D) 1 y 4 Copyright 017 by Sharon Serrago, Inc. Page 105

6 ( )( 3) y Eplanation Begin by factoring each epression that can be factored: y 4( 3) Now cancel 3, as well as a y: ( ) 1 y 4. Writing the result as one fraction makes the answer, choice A. 4y To divide rational epressions, multiply the first epression by the reciprocal of the second epression. Eample 3 Simplify 3 1 C) B) D) 1. (3 1) 4 Eplanation First, change the problem to multiplication by multiplying by the reciprocal of the 9 1 second epression: Then factor where necessary:. In this particular problem, all the factors cancel, so the answer is 1, choice D. To add or subtract rational epressions: Factor each denominator and find the least common denominator (LCD). Multiply the numerator and denominator by the factor(s) of the LCD that the denominator lacks. When both rational epressions have a common denominator, add or subtract the numerators. Don t forget to keep the denominator. Eample 4 Simplify B) C) D) Copyright 017 by Sharon Serrago, Inc. Page 106

7 Eplanation Start by factoring the two denominators to find the LCD: LCD is 35 5or The 3( 5) 5( 5). Multiply both the numerator and denominator of the first fraction by 5, and the second fraction by 3 in the same manner:. 3( 5) 5 5( 5) Distribute the numerators and simplify the denominators:. 15( 5) 15( 5) Now combine like terms in the numerator. Remember to subtract! , answer choice C. 15( 5) 15( 5) 15( 5) A comple fraction is a fraction whose numerator and/or denominator has a fraction in it. These must be simplified. Eample 5 If >, which of the following is equivalent to ? 1 B) 1 1 C) 1 D) Eplanation Find the least common denominator for one denominator of the big fraction: ( )( + 1). Then rewrite the denominator as a single fraction: = 1 1 ( )( 1) = 1 1 ( )( 1) To eliminate the comple fraction, multiply the numerator by the reciprocal of the denominator: 1 1 ( )( 1) The answer is B. Copyright 017 by Sharon Serrago, Inc. Page 107

8 Two polynomials can be divided using the same technique of long division used in arithmetic. Eample 6 Divide: Eplanation Write as a long division problem: term of the divisor and dividend: Start by dividing only the first divided by. This answer goes above the problem. Note: Make sure like powers line up vertically. If a power is missing, use 0 to hold the place. Then multiply back by 4, and write the answer under the dividend: Net, subtract the columns and bring down the net term: Repeat the process by dividing into 3 and follow the same procedure. The quotient and remainder are Synthetic Division could also be used to find the quotient. Copyright 017 by Sharon Serrago, Inc. Page 108

9 For synthetic division, list the coefficients of the dividend: Since the divisor is + 4, we can use = 4 in the bo The results on the bottom line indicate a quotient of 1 + 3, or + 3, with a remainder of 3, just as in the first eplanation. One additional note If a problem asks only for the remainder in a division problem, you can use an even faster 715 technique. For eample, in the previous problem, to find only the remainder for, 4 We could take the dividend ( 7 15) and substitute = 4: (4) 7(4) Notice that this is the same remainder we obtained previously. Eample 7 Divide: B) C) D) Copyright 017 by Sharon Serrago, Inc. Page 109

10 Eplanation Using long division, divide the numerator by the denominator as in eample 5: Therefore, the answer is 3, choice B. 3 7 PRACTICE 3 1. Simplify: 6 1 B) 1 C) 1 3 D) 4y 0. Simplify: y 6y0 y 6y1 B) y 3 C) y D) 1 y 1 Copyright 017 by Sharon Serrago, Inc. Page 110

11 3. Simplify: c 10 c 5 3c6 6c1 4 B) 4 C) 4 c c D) 4 c 5 c 4. Find the quotient and remainder: 54 C) 6 1 B) 1 D) Find the quotient and remainder: C) 7 3 B) D) Copyright 017 by Sharon Serrago, Inc. Page 111

12 6. Find the remainder when is divided by r r 10r r 4 7 B) 7 C) 4 D) 7. If > 0, which of the following epressions is equivalent to y y B) y y C) + y D) y + y y? 8. Simplify: B) 7 10 ( 6)( ) C) 10 ( )( 6) D) Copyright 017 by Sharon Serrago, Inc. Page 11

13 3 9. Simplify: B) 3 ( )( ) C) 6 ( )( ) D) ( )( + )( 3) 10. If > 4, which of the following epressions is equivalent to B) C) 6 8 D) ? Copyright 017 by Sharon Serrago, Inc. Page 113

14 11. The epression is equivalent to which of the following? B) C) D) m 1. When is divided by 3 1, the result is (3, 1) where m is a constant. What is the value of m? ANSWERS 1. A. C 3. C 4. A 5. B 6. B 7. B 8. B 9. C 10. A 11. D 1. 9 Copyright 017 by Sharon Serrago, Inc. Page 114

15 Triangles You will have several problems dealing with triangles on the test. While you already know that a triangle is a 3-sided figure, you should also know that the sum of the measures of the angles in any triangle is 180. In this section we will cover the various kinds of triangles that you are likely to see on the test. Equilateral Triangles An equilateral triangle has all three sides equal. The angles within the triangle are also all equal. Since the sum of all angles in a triangle equal 180, each of the three angles in an equilateral triangle equals 60. b A c Sides a, b, and c are all equal. Angles A, B, and C are all equal (60 each). C a B Eample 1 MOPL is a square and triangle MNO is equilateral. If OP = 9, what is the perimeter of LMNOP? 18 B) 7 C) 45 D) 81 E) 7 M N O Eplanation Since OP = 9, all sides of the square are 9. Since MO = MN = NO, each of these sides is 9 as well. Therefore, the perimeter of LMNOP = = 45, choice C. L P Isosceles Triangles A An isosceles triangle has at least two sides that are equal in length. The third side is not necessarily equal. The measure of the angles opposite the two equal sides are also equal. The altitude of the triangle bisects the base and forms two right triangles. c b Sides b and c are equal in length. Angles B and C are equal in measure. B a C Copyright 017 by Sharon Serrago, Inc. Page 68

16 Eample Triangle SAT is isosceles with SA = AT and = 70. If = 30, what is y? Eplanation S A z y T Since triangle SAT is isosceles, m S = m T = 70. Since a triangle has 180 degrees, m A = m S - m T, so m A = 40. Since = 30, z = = 110. If z = 110, then y = = 70. A scalene triangle is a triangle in which no sides are equal in length. Since all three sides are different lengths, it follows that all angles must be different as well. Right Triangles A right triangle, as the name suggests, has a right angle in it. Since the sum of all angles in a triangle equals 180, and a right angle measures 90, the two remaining angles in the right triangle must have a sum of 90. They are complementary angles. y = 90 + z = 90 y z Pythagorean Theorem Pythagoras was a great mathematician who found that in a right triangle, the square of the longest side of the triangle, called the hypotenuse, is equal to the sum of the squares of the lengths of the other two sides, called the legs. You will see this equation often: a + b = c. It is called the Pythagorean Theorem. If you know the lengths of any two sides of a right triangle, you can find the length of the third side by using this equation. Copyright 017 by Sharon Serrago, Inc. Page 69

17 Eample 3 Find the value of the hypotenuse c in the triangle shown. Eplanation Since the Pythagorean Theorem states that a + b = c, we know the following: (6) + (8) = c = c 100 = c 10 = c 6 8 c Triangles A right triangle whose angles measure have special properties regarding the lengths of the sides. In such a triangle, the length of the sides have a ratio of 1 : 3. Remember that in a triangle, the smallest side is opposite the smallest angle. Therefore, in a triangle, the shortest leg is opposite the 30 angle. 30 long leg = short leg 3 hypotenuse = short leg 60 short leg Eample 4 In the figure drawn, the hypotenuse of a triangle is 1. What are the values of and y? y 30 1 = 3, y = 6 B) = 6, y = 6 C) = 6, y = 6 3 D) = 6 3, y = 6 60 Eplanation The short leg,, is ½ 1 = 6. The longer leg, y, is 3 short leg, or 3 y = 3 6 = 6 3 y = 6 3 The correct choice is C. Copyright 017 by Sharon Serrago, Inc. Page 70

18 Eample 5 In the triangle shown, the length of the short leg is 8 3. What are the lengths of and y? 8 3 y 30 = 16, y = 4 B) = 16 3, y = 4 C) = 4, y = 16 3 D) = 4, y = 4 Eplanation 8 3 is the short leg. Therefore, the hypotenuse, y, is 8 3, or The longer leg,, is = 8 3 = 4. y = 16 3 = 4 The correct choice is C. The rules for the triangle are commonly applied to an equilateral triangle. If we take two triangles and set them back to back, the long leg of the triangle becomes the height of an equilateral triangle. You might have problems on the PSAT/SAT in which you must split an equilateral triangle into right triangles in order to solve the problem Eample 6 The perimeter of an equilateral triangle is 4. What is the height of the triangle? 4 B) 4 3 C) 8 D) 8 3 Eplanation Since the perimeter is 4, each side is 8. The height of the equilateral triangle forms two triangles. Since the height also divides the base in half, this forms the triangle shown at right Therefore, h = short leg 3 h = 4 3, choice B. Copyright 017 by Sharon Serrago, Inc. Page 71

19 Triangles An isosceles right triangle is also known as a triangle. Since the two acute angles are equal in measure, the two legs are equal in length as well. In a triangle, the length of the sides have a ratio of 1:1: hypotenuse = It is helpful to know that the diagonal of a square divides the square into two triangles in which the diagonal is the hypotenuse = Eample 7 From the figure at the right, find the value of. 45 B) C) 4 D) Eplanation is the hypotenuse of the triangle. To find the hypotenuse, multiply 4 by. = 4, choice D. Copyright 017 by Sharon Serrago, Inc. Page 7

20 Eample 8 A Given ABC at the right, find the length of AC. 5 5 B) 5 C) 5 D) 10 C B Eplanation This time we must work backwards to find the length of a leg. We divide the hypotenuse value 5 by : AC = 5 = 5 AC = 5, choice B. Eample 9 Find the diagonal of a square with a perimeter of B) 10 C) 0 D) 0 Eplanation If the perimeter is 40, then each side = 40 = The diagonal cuts the square into two triangles. Now the diagonal is the hypotenuse of a triangle. Thus, the diagonal = 10 = 10 = 0, choice C Copyright 017 by Sharon Serrago, Inc. Page 73

21 Pythagorean Triples You have already seen the Pythagorean Theorem in the previous pages. There are some sets of numbers describing the side lengths of a right triangle that are very common in mathematics. We refer to these common sets as Pythagorean Triples Pythagorean Triples If you multiply one of these triples by a constant, you get another triple. For eample, multiplying by 4 gives There are an infinite number of Pythagorean triples, but the four to the left are the most important. These triples MUST be MEMORIZED! Eample 10 From the given figure at the right, find the value of. Eplanation Note that both 9 and 15 are divisible by 3. Thus, and What triple do you know 3 that has one leg of length 3 and a hypotenuse of 5? 3-4-5! Multiplying by 3 gives us a new triple: Therefore, = Eample From the figure at the right, determine the length of y. Eplanation Figure not drawn to scale The smaller triangle is a triangle, so = 5. That gives the larger triangle a leg of 5 and a hypotenuse of 13. This is the triple. Therefore, y = y Copyright 017 by Sharon Serrago, Inc. Page 74

22 Triangle Inequality In any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Look at the given triangle with sides a, b, and c. According to the triangle inequality, all three of the following must be true: a+b > c a+c > b b+c > a a b c An easy way to get the range of values for a missing side is to find both the sum and the difference of the two given sides. The third side must be between these two numbers. Eample 1 If a triangle has two sides of lengths 9 and 4, what is a possible length for the third side? Eplanation Let the missing side =. Find the sum and the difference of the two given sides: = = 5 The third side must be between 5 and 13 or 5 < < 13 Therefore, can be any value larger than 5 and smaller than 13. PRACTICE 1. Find the length of the longest leg of a triangle if the hypotenuse is 8. 8 B) 4 3 C) 16 D) 8 3. In the figure, WXYZ is a rectangle. C is the midpoint of WX, and D is the midpoint of XY. What is the perimeter of WCDYZ if WX = 10 and XY = 4? X C D Y 64 B) 47 C) 68 D) 30 W Z Copyright 017 by Sharon Serrago, Inc. Page 75

23 3. Find the length of DE. 4 B) 4 3 C) 4 6 D) 8 D 60 F 8 E 4. In the given figure, XY = 3. What is the value of WZ YZ? 3 1 B) 6 C) 3 6 D) W 45 Z X 60 Y 5. In the given figure R is a right angle. MQ = QP and m S = 38. What is the measure of M? 59 B) 6 C) 5 D) 64 M R P Q S 6. In the given triangle, MN = MP = PL, and m LMP = 6. Find the value of c. M c 76 B) 64 C) 5 D) 78 L P N Copyright 017 by Sharon Serrago, Inc. Page 76

24 7. ABC is equilateral. E is the center of ABC. If DC = 9, what is the length of BE? A D 3 3 B) 6 C) 6 3 D) 4 3 B E C 8. Find the length of the hypotenuse of a triangle if the shorter sides are each B) 4 C) 4 5 D) Which of the following could not be the lengths of the sides of a triangle? 6, 8, 1 B) 7, 8, 14 C), 4, 5 D) 3, 5, The lengths of two sides of a triangle are 3 and 17. What is one possible length of the third side? Copyright 017 by Sharon Serrago, Inc. Page 77

25 11. A ship sailed 60 km and then 47 km on a perpendicular course. The ship will return to its starting point by the shortest route. Find the total distance traveled to the nearest kilometer. 1. In the figure, AB = BD. What is the value of a? C 61⁰ B) 67⁰ C) 75⁰ D) 79⁰ A B 11 E a 33 D 13. In the figure, BD and AC intersect at point E, AE = ED, and BE = CE. What is the measure (in degrees) of ACB? )Disregard the degree symbol when gridding your answer.) B 50 E A C 80 D ANSWERS 1. B. A 3. C 4. D 5. D 6. A 7. B 8. A 9. D < < B Copyright 017 by Sharon Serrago, Inc. Page 78

26 Congruence and Similarity Congruent and Similar Triangles Triangles are said to be congruent if they have the same size and same shape. If two triangles are congruent, then the corresponding sides of the triangles have the same length, and corresponding angles have the same measure. Given ABC is congruent to DEF below: E B A C D F All corresponding angles are equal: m A m D m B m E m C m F All corresponding sides are equal: AB DE BC EF AC DF Eample 1 TUV and XYZ are congruent. What is the perimeter of XYZ? T Eplanation 8 10 Since the two triangles are congruent, TV = XZ = 10 TU = XY = 8 UV = YZ = 4 U V Z 4 Y X Perimeter of XYZ is = Copyright 017 by Sharon Serrago, Inc. Page 79

27 Eample Given QRS TUV. If mr 11 1 and mu 9 5, find m U. 3 B) 1 C) 7 D) 3 Eplanation Since R and U are corresponding parts of congruent triangles, they are congruent. Set their values equal to each other: Solving for yields 3. The question asks for the value of U, so substitute 3 back into the epression 9 5: 9(3) The correct answer is D. Similar triangles have the same shape, but are not necessarily the same size. In similar triangles, all corresponding angles are equal in measure, and all corresponding sides have the same ratio. Given ABC is similar to JKL: B K A C J L All corresponding angles are equal: m A m J m B m K m C m L All corresponding sides have the same ratio: AB BC AC JK KL JL Eample 3 Find the value of a if ABC is similar to XYZ. Eplanation A B Since the two triangles are similar, the following proportion is true: BC YZ = AB a a XY = Therefore, = or a = a C X 11 Y 8 Z Copyright 017 by Sharon Serrago, Inc. Page 80

28 Proportional Segments In a triangle, if a line segment through the triangle is parallel to a side of the triangle, the line divides the other two sides proportionally. Eample 4 Find the value of. 5 B) 6 C) D) 37 6 Eplanation Since the middle segment is parallel to the third side of the triangle, the two other sides of the triangle are in proportion: Net, cross multiply to solve: Therefore,, 5 6 choice C. The same rule applies when parallel lines intersect two transversal lines the segments are divided proportionally. Eample 5 Find the value of a. B) 3 C) 6 D) 18 a+3 a 6 18 Eplanation Since the segments are divided proportionally by the parallel lines, set up a proportion: Cross multiplying and solving for a yields a 3, answer choice B. a 6. a 3 18 Copyright 017 by Sharon Serrago, Inc. Page 81

29 Two triangles can be proven similar in several different ways, but the most common is the Angle Angle Similarity Postulate. If two angles of one triangle are equal to two corresponding angles of a second triangle, then the two triangles are similar. Note: the symbol for similar is ~. A D C y B F y E In the two triangles above, since mb meand mc m F, ABC ~ DEF. Since the two triangles are similar, AB BC AC. DE EF DF Eample 6 Find the value of m. 4 5 B) 4 7 C) 4 D) m Eplanation The two parallel line segments form two pairs of alternate interior angles. Additionally, the diagram has a pair of vertical angles. Therefore, the two triangles are similar by the AA Similarity Postulate (even though there are 3 pairs of congruent angles, only pairs are needed). Be careful when setting up the proportion some students might pair the side of length 4 with the side of length 5, but the side of 4 corresponds to the side of length 7 if you pair up the congruent angles. The proportion becomes Solve for m to get, answer choice B. m 4 7 Copyright 017 by Sharon Serrago, Inc. Page 8

30 In a right triangle, the altitude to the hypotenuse forms two right triangles that are both similar to the original triangle. In the picture at the right, ACB ADC CDB. C Since the corresponding sides of these triangles are in proportion, it follows that z. z y A z D y B Eample 7 Find the value of. 4 3 B) 6 C) 6 3 D) Eplanation Using the ratio above between the two smaller triangles, the following proportion can be solved: Therefore, 48, and The correct answer is A. PRACTICE 1. Find the value of B) 35 C) 6 D) 35 7 Copyright 017 by Sharon Serrago, Inc. Page 83

31 . Find the value of. 6 6 B) 8 C) 1 5 D) Find the value of a. 5 3 B) C) 8 5 D) a 5 8 a+3 4. Given that, find the value of a. 0.5 B) 1 C) D).5 a + a a + 5 a + 1 p q r 5. Given that, find the value of. 4 3 y + 4 B) 4 13 C) 5 6 D) 4 l 3y+1 m - 1 n 5 q Copyright 017 by Sharon Serrago, Inc. Page 84

32 6. Find the value of. B) 3 C) 4 D) In the figure below, the ratio of MO to NO is : 3. If NP 4, what is the length of ML? 6 B) 1 C) 16 L M O N D) 18 P 8. In the figure, XYZ BAZ. Which of the following must be true? YXZ BAZ B) AZ = ZY C) D) A X Y Z B Figure not drawn to scale 9. In the figure below,, and What is the length of segment? L 4 M N 3 O P Copyright 017 by Sharon Serrago, Inc. Page 85

33 10. QRS is similar to WXY. R Find the length of side a S 7 B) 4 C) 8 Q 1 Y X a W D) If triangles BCD and FGH are congruent, what is the sum of their perimeters? 6 B) 40 C) 46 D) 5 B D 11 9 C F 6 H G 1. Given the two triangles below, what is the value of? 1 3 B) 15 C) 39 D) E C In the triangle above, AB, CD, and EF are parallel, and the height of the entire triangle is 0 units. Find the height h between AB and CD. 5 A h B D F 0 Copyright 017 by Sharon Serrago, Inc. Page 86

34 14. In the figure, RV is parallel to US. What is the length of RT? R 10 S 5 V U 1 T 15. X A B 7 Z Y 63 C Triangles XYZ and ABC are shown above. Which of the following is equal to the ratio of YZ XZ? BC AB B) AB BC C) BC AC D) AC BC ANSWERS 1. A. D 3. D 4. B 5. A 6. D 7. C 8. D 9. 5/3 10. C 11. D 1. B Copyright 017 by Sharon Serrago, Inc. Page 87