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ACT Review Chapter One Start Here Overview of PowerPoint This PowerPoint will cover: ◦ ◦ ◦ ◦ ◦ ◦ ACT is predictable How to prepare What is the ACT How is the ACT scored? Dates to Take ACT vs. SAT ACT and Predictability The ACT tests the same way, year after year. There are always 14 plane geometry questions and 10 questions on pronunciation. So don’t stress – this course will get you ready for what is ahead. And feel free to talk to others who have taken it – their experiences may help you. How (and why) To Prepare Your ACT score is just a part of the whole academic package you present to a college when you apply for admissions Schools look at test scores, grades, extracurricular activities, essays, and recommendations. Your ACT score is the EASIEST of these to change. What is the ACT? Time: about three and a half hours with one break. English: 45 minutes – 75 questions Math: 60 minutes – 60 questions Reading: 35 minutes – 40 questions Science Reasoning: 35 minutes – 40 questions Optional Writing Test: 30 minutes How is the ACT Scored? Scores for each of the four tests are graded on a scale of 1 to 36. The four scores are averaged to give you a composite score. Next to each score is a percentile ranking comparing you with all of the other testtakers. How is the ACT Scored..con’t Scores have subcategories ◦ These subcategories are there for your benefit ◦ They will tell you where you are the weakest – therefore study those and do better next time! ◦ Writing has two grades – one is 1 to 36 and the other is 2 to 12. Neither score contributes to your composite score (and it costs an extra $14.50) Dates to Take It… The test is given five times each year, usually at 8:00 a.m. on a Saturday morning. They may start testing on school days, but that is still in the works. You need to go to www.act.org to determine which administrations of the test are best for you. How to Take It? You need to obtain a registration packet – ◦ Get one from your guidance counselor’s office ◦ Online from www.actstudent.org/forms/stud_req.html ◦ Write or call ACT ACT Registration P.O. Box 414 Iowa City, IA 52243 319-337-1270 ACT vs. SAT Princeton Review believes that the SAT is not nearly as fair as the ACT ACT measures achievement, not ability SAT is less time-pressured than the ACT Many questions on the SAT are ‘trickier’ than the ACT SAT Verbal has a stronger emphasis on vocabulary SAT Math section mainly tests algebra and plane geometry and has no trigonometry questions. Both tests have an essay section Chapter Two Triage Intro The ACT is a timed-test You can’t waste any time on questions A good way to not waste time is to see which questions are easier than others – triage. Test Organization The ACT questions are not organized from easy to difficult – so as you take the test, the questions will not get progressively difficult. Remember that an easy question counts the same as a difficult one – do the easy ones first! What is Triage? Triage is a method hospitals use in emergency rooms to see who needs to receive medical attention first. You need to learn to do the same with the test – if you glimpse at a question and it appears to be really difficult, move on – do the easy ones first and then return to the really hard questions. How to Triage When you come to a question, be thinking ‘now, later, never.’ If it is easy – do it now. If it is difficult, but you might figure it out, do it later. If it is utterly impossible, guess and move on – there are plenty of other questions to attack. Making Passes Princeton Review recommends making two passes on each section. On the first pass, answer every single question you can answer. That way you never run out of time on the ones you know how to do. If you get stuck on a problem on the first pass, circle it and move on. You Didn’t Waste Time If you spent time on a question but circled it and moved on, you did not waste your time. When you make the second pass, you will already know a little about the question and thereby making it easier for you to answer. Maybe by revisiting the question, you will see something you missed or realize how to solve the problem. Second Pass After you finish the first pass (answered easy questions, circled difficult questions, guessed at impossible questions, filled in the letter of the day), look over the remaining problems and ask ‘which one do I want to do now?’ On the second pass, you have time to attack the questions without worrying about running out of time. Why? Because you already answered everything. How Will Triage Help Me? If you use triage – asking yourself do I answer the question now, later or never – you will spend less time on easy and impossible questions and more time on the questions you have a fighting chance to answer. Do this and you will score more points. The End Chapter Three Guessing and POE Table of Contents No Penalty POE Letter of the Day No Penalty When in doubt…GUESS! There are 215 questions on the ACT You get no points for answers left blank There is no penalty for guessing – in other words you should never leave a question unanswered. POE POE stands for Process of Elimination. If you do not understand a question, start with the answers and work backwards. POE Example Look at this question: 1. What is the capital of Myanmar? ◦ ◦ ◦ ◦ A. B. C. D. Bangkok Even though you may not know the Tokyo capital of Myanmar, you can probably cancel out two of the possible answers – YangonTokyo and Berlin – because these are recognizable. The others may Djiboutimore seem a little more difficult. But you have reduced the problem to a 50/50 guess which is much better than guessing at random. Letter of the Day If you guess using the same letter every time, you will pick up more points than if you were guessing at random. You will not get some of your random guesses correct while using the same letter – however, if you change your guess answer, you might miss all of them. Letter of the Day, cont. It doesn’t matter what letter you choose as the letter of the day – just be consistent. Chapter Four Taking the ACT Contents of this PowerPoint Preparing for the ACT ◦ Night before, day of, warming up Things to remember Art of Test Taking Keep Your Mouth Shut… Watch Where You Bubble Should I Cancel My Scores?? Prepare Take this course Read the book Look into different techniques Review Take full practice tests Get a good night’s rest Eat breakfast Bring a snack to the test Don’t chat during the break – focus and eat your snack – you need the energy and the concentration Don’t Leave Home Without It You will need these items on the test day: ◦ ◦ ◦ ◦ ◦ Admissions ticket Photo ID or letter of identification Plenty of sharpened #2 pencils A watch An acceptable calculator with new batteries Things to Remember Know where the test center is Show up early Lay out what you need the night before so you don’t leave anything at home Bring a snack and a bottle of water Art of Test Taking Concentrate Ignore outside noises Once one test ends, forget it – the most important thing is what you are taking at that moment. Don’t worry about previous tests and answers because that will slow you down and not allow you to concentrate. Keep Your Mouth Shut… Don’t let anyone cheat off of your paper. The test people are really smart and will catch you if that is what you try to do. If your answers are similar to those around you, the computer will notice and the guilty and not guilty will be invited to take the test over. Watch Where You Bubble Make sure you bubble in your answers Make sure when you are looking at the test booklet and then go to bubble in your answer that you are filling in the bubble on the write number. There is nothing worse than writing the correct answer down in the wrong blank – because you get them both wrong. Watch out… con’t Try to write your answers in the test booklet and on your answer sheet. That way when you finish, you can go back and double check you didn’t make any careless mistakes. If you get your scores back and they seem out of line, you can ask the ACT examiners to review your answer sheet for ‘gridding errors’. But don’t rely on this – do it right the first time and don’t make a mistake. Should I Cancel My Scores?? If you are wise, you can make sure your scores do not go to a college until you are sure it is a good test. Princeton Review does not recommend canceling scores. When you take the ACT more than once, you can choose which score you want to report. Chapter Five Introduction to the ACT English Test Contents What the test tests… What your score means Example passages Triage Look for Clues What if there is more than one error? POE, No Change and OMIT Basic Terminology What the Test Tests The ACT English test measures how well you understand standard English You have five passages to read and parts of them will be underlined. You must determine if the parts are correct or incorrect. You will have 45 minutes to answer 75 questions. What your score means It is difficult to test language proficiency on a test. A good score or a bad score doesn’t necessarily represent a true picture of your writing ability. Just prepare for the exam and do your best Example Passages On pages 32 and 33 of your textbook, you will find examples of ACT English Test passages. These are typical and we recommend that you practice taking full-length practice tests in order to get proficient in taking this style of test. You are allotted 36 seconds per question – so practice will not only help you get used to the style of question, but hopefully help you get quicker at finding the answer. Triage Don’t forget to make two passes. Don’t waste time If you aren’t quite sure of the answer, use POE. If it still isn’t clear, guess, circle the question and move on. Don’t guess too quickly though and don’t just pick the first answer that ‘sounds right’ – the English that we hear in the hallways at school isn’t always grammatically correct. Look for Clues If you are not sure of the answer, focus on the answers. If you notice that all of the answers have pronouns, the question most likely is asking about pronouns and/or pronoun usage. What if there is more than one error? If there is more than one error in the underlined portion of the sentence, don’t freak out. This happens often on the test. First, find one error and then eliminate the answer choices that contain the same error. Then look at the remaining answer choices. Don’t worry about the number of errors you find in the question – focus on the differences in the answer choices. POE, No Change and OMIT Sometimes you will not know the right answer, but you definitely know two are wrong. Use this to your advantage. Cross out the answers that you know are wrong and look at the question again. If you still don’t figure it out, circle the question, cross out wrong answers, guess now and come back to it later. Guess right then because you don’t want to bubble in the letter of the day if that letter you know is wrong for that particular question. POE, No Change and OMIT No change is the correct answer approximately 1 out of 4 times. Don’t be afraid to use NO CHANGE because there is a 25% chance it is the correct answer. POE, No Change and OMIT A few of the questions will give you the option to omit the underlined portion. When this is offered, there is a high chance that it is the right answer – sometimes more than 50% of the time it is the correct answer. Basic Terminology Basic Terms you need to know: ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ Noun Verb If you do not know these terms, you need Article to do two things: Adverb 1. Apologize to your English Adjective teachers for not listening Preposition in class. Phrase 2. Find an English grammar Prepositional Phrase book and review the Pronoun basics. Clause Chapter Six Sentence Structure and Punctuation The Basics A sentence must have three things: ◦ A subject ◦ A verb (predicate) ◦ A complete thought Basics, cont. A clause has a subject and a verb A phrase is missing at least one of these Commonly Used Sentence Structures Independent Clause. New independent clause. Independent Clause, (conjunction) independent clause Independent clause; independent clause. Independent clause, dependent clause. Dependent clause, independent clause. Independent Clause. New independent clause Example: ◦ John went to school. His mom went to work. Independent Clause, (conjunction) independent clause. John went to school, and his mom went to work. When two independent clauses appear in the same sentence, they are usually joined together by a conjunction (i.e. and, or, but, for, nor, or yet). Independent Clause, (conjunction) independent clause. Con’t… A comma goes before the conjunction that joins the two independent clauses. Independent clause; independent clause. John went to school; his mom went to work. If two independent clauses are separated with a semicolon, it should not have a conjunction immediately after the semicolon. Two independent clauses can be separated with a colon if the second clause is an extension of the first – “I didn’t know where to go to school: I could either go to Lanier High School or transfer to St. Joseph’s. Independent clause, dependent clause. John went to school, while his mother went to work. Commas are used to separate these two. A dependent clause depends on the other clause, therefore it cannot stand alone. Dependent clause, independent clause. When John went to school, his mom went to work. Independent and Dependent Clauses Commas separate independent clauses from dependent clauses. If a clause cannot stand on its own, then it is a dependent clause. Independent Clause and a Modifying Phrase Tired and upset, she flopped down on the couch and immediately went to sleep. The modifying phrase, “Tired and upset”, must be separated from the independent clause with a comma. You just have more to say… If you want to give more information in an independent clause, you can use a colon. I had to make out my schedule for college, but I could not decide. I wanted to enroll in many classes: Biology 101, English 200, Russian 201 and Physics 114. Commas: restrictive and nonrestrictive clauses. Commas can change a restrictive clause or phrase into being nonrestrictive. A restrictive clause or phrase is essential to the meaning of the sentence. A nonrestrictive clause or phrase is not essential to the meaning of the sentence. Commas: Examples Restrictive clause/phrase: People who always want more money and possessions should place more emphasis on other people. “People who always want more money and possessions” is necessary because it explains ‘who’ should place more emphasis on other people. Commas: Examples Nonrestrictive clause/phrase: My friend, who always gripes that he doesn’t have enough money, can always be found on the couch playing the PlayStation. “Who always gripes that he doesn’t have enough money” can be deleted because it just adds extra information. Meaning will not be lost if it is deleted. Mr. Hyphen and Mrs. Dash A hyphen is used to divide words that are cut in half at the end of the line or to join two words together. Example: Man-eating tiger found my paleontologist. A dash is a longer line used to bring emphasis to part of the sentence. Example: Jonathan – who is crazy as a loon – leapt off the bridge into the swirling, turbulent waters below. How to find the dash error If a question on the test has an underlined portion or any of the answer choices contains a dash, compare the dash to the punctuation marks available in the answer choices. If the sentence contains a sudden break in thought, an afterthought, or an explanation, then a dash is appropriate. If the highlighted portion of the sentence is in the middle, then it needs two dashes. If it is a the end, then it only needs one dash. My colon has a problem If one of the answer choices has a colon or the question has an underlined phrase with a colon, check if it introduces a list. Also, make sure that the sentence contains an independent clause. One of the ACT’s tricks is to have a nice colon and list, but to have it follow a dependent clause. Bad things, man. Really bad things. Do not, I repeat, do not make these simple, careless errors: ◦ ◦ ◦ ◦ ◦ Sentence fragments Run-ons Comma splices Misplaced modifiers Non-parallel construction Sentence Fragments If the sentence does not have a subject, verb and complete thought, then it is not a sentence. It is a phrase or a dependent clause. The words as, although, because, despite, if, how, that, what, when, where, who, why, or while can easily change an independent clause into a dependent clause. Example: John went to class. (independent) ◦ When John went to class. (fragment (dependent)) It takes two kinds… Fragment One: a dependent clause waiting for its other half that is not there. Example: As the bus drove by, honking and swerving. After the comma, there should be an independent clause, but there is none. It takes two kinds… Fragment Two: A fragment that you need to add to an existing independent clause before or after it. Example: The scientist’s plan to reverse the effects of aging through biotechnology included the use of synthetic cellular structures if he is successful, he will apply for numerous patents on his creation. On the test you will be given something similar to the following answer choices: F) NO CHANGE G) structures; if, H) structures. If I) structures, if Correct answer on the next slide… The Answer is… H) structures. If Look at the sentence with the revision written in: The scientist’s plan to reverse the effects of aging through biotechnology included the use of synthetic cellular structures. If he is successful, he will apply for numerous patents on his creation. H would have been a good choice except for the comma placed after if. G doesn’t help because it keeps the sentence as a run-on. F doesn’t work because it is a run-on as well. Misplaced Modifiers A modifying phrase needs to be near what it is modifying. If it is too far from what it modifies, it can get misplaced. Example: Jenny spent three hours getting ready for prom. Coming down the stairs in an elegant, blue evening gown, her father exclaimed, “Jenny, you look great!” Do you see how it reads like Jenny’s father is wearing the evening gown? Modifying Phrases If a sentence begins with a modifying phrase – just like in the example – it needs to be followed by the noun that it modifies. How to look out for these? When you see a modifying phrase followed by a comma, make sure the noun that immediately follows is what needs to be modified. Misplaced Modifier’s Cousin A Construction Shift is very similar to a misplaced modifier – except that the does not need any words changed, rather they need to be moved. Example: Stepping to avoid the broken steps, the salesman carefully fell off the front porch and into the bushes. ◦ ◦ ◦ ◦ A.NO CHANGE B. (Place after Stepping) C. (Place after and) D.(Place after fell) And the answer is… Place after stepping. Why? Because a person does not carefully fall down, but a person can step carefully. Non-Parallel Construction Hugged, gave, cried Smart, handsome, polite Honesty, integrity, conviction The previous lists of three are all parallel – in other words, they have the same part of speech (verb, adjective, noun – respectively) Non-parallel, cont. Look at the following question: To taste Aunt Susie’s chocolate pie is experiencing perfection. This has a problem because the verbs are not parallel – ‘to taste’ and ‘is experiencing’ Look for an answer that either changes ‘to taste’ to ‘Tasting’ or changes ‘is experiencing’ to ‘to experience.’ Punctuation Problems: Apostrophes An apostrophe shows possession or to mark missing letters in a word (contractions). Possession: Martha’s Vineyard, Elvis’s Cadillac, Achilles’ Heel, John’s truck Contractions: Don’t, Needn’t, Would’ve, Isn’t You better know it: It’s, Its’, Its It’s = It is Its’ = Is not a word Its = possessive of its References http://owlet.letu.edu/grammarlinks/punctu ation/punct4d.html Chapter Ten Identifying Sentence Errors Approaches 1. 2. 3. Read each sentence quickly but carefully Consider each question to be independent of the others Think about how the sentence would sound if you were reading them aloud – you may ‘hear’ the difference even when you can’t ‘see’ it. Approaches, cont 4. 5. 6. 7. Look at all of the answer choices. Initially look for the most common errors – if errors are the most common, it means they are easy to make and therefore makes the test makers that you too can make the error. Know your idiomatic phrases Know that some sentences don’t have any errors Approaches, cont 8. 9. When you take practice tests, try to correct the error when you find them – even when it just wants you to find the error. This helps you keep in mind what correct sentence looks and sounds like. Move quickly through the questions – if you can save time looking for sentence errors, you can spend more time on paragraphs and essays Approaches, cont If you skip a question, mark it. Mark through the answers that you know are wrong. Participles Participles are verbs that are used as adjectives. For example: ◦ The falling leaves reminded me that winter was coming. To fall is a verb, but in this sentence it is used as an adjective (falling) to modify leaves. ◦ The prepared meal was getting cold because no one wanted to sit down at the table.To prepare is a verb but in this sentence it is used to modify meal. Excuse me, your participle is dangling… Look at the following sentences: I thought I heard a strange noise walking through the woods. The cook cooked the lobster wearing a funny hat and apron. Do you see how the modifying phrases need to be moved to the nouns they modify? (walking through the woods and wearing a funny hat and apron). They should read: Walking through the woods, I thought I heard a strange noise. The cook wearing a funny hat and apron cooked the lobster. Chapter Seventeen Geometry and Measurement Review Concepts You Need to Know Geometric notation Points and lines Angles in a plane Isosceles triangles Equilateral triangles Right triangles/ Pythagorean theorem 30-60-90 triangles 45-45-90 triangles 3-4-5 triangles Congruent triangles Similar triangles The triangle inequality Parallelograms Rectangles Squares Areas of squares and rectangles Perimeters of squares and rectangles Area of triangles Area of parallelograms Angles in a polygon Polygon Perimeter Polygon Area Circle Diameter Circle Radius Arc Tangent to a circle Circumference Area of a circle Solid geometry Surface area Volume Geometric perception Coordinate geometry Slopes, parallel lines Perpendicular lines Midpoint formula Distance formula transformations Geometric Notation Notation means the symbols used to express certain things – in geometry, different notation is used to express types of lines, angles and shapes. For more on notation, visit http://www.beva.org/math323/asgn1/sep5.htm For a video on geometric notation: http://www.youtube.com/watch?v=BhukE0gRE5Q &feature=player_embedded Points and Lines If you are given a line (f) and it has two points, A and B. The midpoint between A and B is Q – so AQ and BQ are equal to each other. Therefore, if AQ=6 then BQ=6 and AB=12. Angles in the Plane Some basic facts about angles formed in a plane by lines, line segments and rays. 1. Vertical angles and supplementary angles: two opposite angles formed by two intersecting lines are called vertical angles. 2. Vertical angles have the same measure. Angles next to each other add up to 180 degrees. Two angles that add up to 180 are called supplementary angles. Parallel lines: if a line intersects a pair of parallel lines, eight angles are formed. 4. Right angle: an angle with a measure of 90 degrees. 5. If two lines intersect and one of the four angles formed is a right angle, then the lines are perpendicular. 6. Two angles whose measures have a sum of 90 degrees are called complementary angles. 3. Triangles (Including Special Triangles) The sum of the measures of the angles in any triangle is 180. Equilateral Triangles The three sides of an equilateral triangle are equal in length. The three angles are also equal, so they each are 60 degrees. Isosceles Triangles An isosceles triangle is a triangle with two equal sides. The angles opposite the two equal sides are equal. Right Triangles and the Pythagorean Theorem A right triangle is a triangle with a right angle (90 degrees) So if one angle is 90 then the other two angles are complimentary (add up to 90 degrees) You can find the lengths of a right triangle sides by using the Pythagorean Theorem: a2+b2=c2 Where c is the hypotenuse. 30-60-90 Triangles 30-60-90 Triangles The guy who named 30-60-90 triangles didn’t have much of an imagination. These triangles have angles of 30,60, and 90. What’s so special about that? This: The side lengths of 30-60-90 triangles always follow a specific pattern. Suppose the short leg, opposite the 30° angle, has length x. Then the hypotenuse has length 2x, and the long leg, opposite the 60° angle, has length x√3. The sides of every 30-60-90 triangle will follow this ratio of 1:√3:2. 45-45-90 Triangles A 45-45-90 triangle is a triangle with two angles of 45° and one right angle. It’s sometimes called an isosceles right triangle, since it’s both isosceles and right. Like the 3060-90 triangle, the lengths of the sides of a 4545-90 triangle also follow a specific pattern. If the legs are of length x (the legs will always be equal), then the hypotenuse has length x√2. 3-4-5 Triangles Because right triangles obey the Pythagorean theorem, only a specific few have side lengths that are all integers. For example, a right triangle with legs of length 3 and 5 has a hypotenuse of length = 5.83. The few sets of three integers that do obey the Pythagorean theorem and can therefore be the lengths of the sides of a right triangle are called Pythagorean triples. Here are some common ones: {3, 4, 5} {5, 12, 13} {7, 24, 25} {8, 15, 17} In addition to these Pythagorean triples, you should also watch out for their multiples. For example, {6, 8, 10} is a Pythagorean triple, since it is a multiple of {3, 4, 5}. The SAT is full of right triangles whose side lengths are Pythagorean triples. Study the ones above and their multiples. Identifying Pythagorean triples will help you cut the amount of time you spend doing calculations. In fact, you may not have to do any calculations if you get these down cold. Congruent Triangles Are identical. Two triangles are congruent if they meet any of the following criteria: All corresponding sides are equal. This is known as the Side-Side-Side (SSS) method of determining congruency The corresponding sides are equal, and the mutual angles between those corresponding sides are also equal. This is known as the Side-Angle-Side (SAS) method of determining congruency The two triangles share two equal corresponding angles and also share any pair of corresponding sides. This is known as the Angle-Side-Angle (ASA) method of determining congruency Similar Triangles Corresponding angles of similar triangles are identical. Corresponding sides of similar triangles are proportional. The Triangle Inequality The length of any side of a triangle will always be less than the sum of the lengths of the other two sides and greater than the difference of the lengths of the other two sides. Quadrilaterals There are relationships among the angles and sides in parallelograms, rectangles, and squares. Parallelograms Opposite angles are equal Opposite sides are equal Rectangles All angles are right angles Squares The lengths of all the sides are equal. Areas and Perimeters Rectangles and Squares • Area=length x width • Area = s2 • The perimeter is the sum of all sides. Or (length + width) times 2. • The perimeter of a square can be found by multiplying 4 times the length of any side. Triangles ◦ Area = (1/2)bh, where b is the base and h is the height. Parallelograms ◦ The area is length x height Other Polygons A regular polygon is a polygon whose sides are all the same length and angles have the same measure. You can find the total number of degrees in the interior angles by dividing the polygon into triangles. Since one triangle has 180 degrees for its interior angles, just multiply the number of triangles times 180. Circles Diameter (D) = 2r Radius (r)= D/2 Length of Arc = D ∏(degrees of angle / 360) Tangent to a Circle Circumference= ∏D Area= ∏ r2 Solid Geometry (lots to remember) Volume of the cube = s3 Surface area of a cube = 6s2 Volume of rectangular solid = lwh Surface area of rectangular solid = 2(lw + wh + lh) Volume of prism = area of base × height Volume of a Cylinder = ∏r2h Volume of a Sphere = 4/3∏r3 Surface Area of a Sphere = 4∏r2 Volume of a Cone = 1/3∏r2h Volume of a Pyramid=1/3 x area of base x height Coordinate Geometry Slopes, Parallel Lines, Perpendicular Lines For a great explanation on coordinate geometry, go to http://www.math.com/school/subject3/less ons/S3U1L2DP.html . Midpoint Formula A midpoint is a point that denotes the middle of any given line segment. The Midpoint Theorem says the x coordinate of the midpoint is the average of the x coordinates of the endpoints and the y coordinate is the average of the y coordinates of the endpoints. If a line segment has the end points (x1, y2) and (x2, y2), the midpoint is given by the following formula: [((x1 + x2)/2), ((y1 + y2)/2)]. Distance Formula The distance formula says that the distance d between any two points with coordinates (x1, y1) and (x2, y2) is given by the following equation: d = SQRT[(x2 - x1)2 + (y2 - y1)2]. Problem: Find the distance between (-2, 3) and (8, -1). Solution: Plug any given information into the distance equation. d = SQRT[(8 - (-2))2 + (-1 - 3)2] Simplify. d = SQRT[102 + (-4)2] d = SQRT(100 + 16) d = 2(SQRT(29)) This information and more can be found at http://library.thinkquest.org/20991/geo/coordgeo.html Transformations A transformation is when you take a figure that is drawn on a plane and then move the figure. Three ways for a figure to ‘transform’ are: ◦ Rotation (turn) ◦ Reflection (flip) ◦ Translation (slide) Remember that the figure retains the same size and shape – you just adjust the points of the figure by moving up or down according to the problem. Chapter Seventeen Geometry and Measurement Review Concepts You Need to Know Geometric notation Points and lines Angles in a plane Isosceles triangles Equilateral triangles Right triangles/ Pythagorean theorem 30-60-90 triangles 45-45-90 triangles 3-4-5 triangles Congruent triangles Similar triangles The triangle inequality Parallelograms Rectangles Squares Areas of squares and rectangles Perimeters of squares and rectangles Area of triangles Area of parallelograms Angles in a polygon Polygon Perimeter Polygon Area Circle Diameter Circle Radius Arc Tangent to a circle Circumference Area of a circle Solid geometry Surface area Volume Geometric perception Coordinate geometry Slopes, parallel lines Perpendicular lines Midpoint formula Distance formula transformations Geometric Notation Notation means the symbols used to express certain things – in geometry, different notation is used to express types of lines, angles and shapes. For more on notation, visit http://www.beva.org/math323/asgn1/sep5.htm For a video on geometric notation: http://www.youtube.com/watch?v=BhukE0gRE5Q &feature=player_embedded Points and Lines If you are given a line (f) and it has two points, A and B. The midpoint between A and B is Q – so AQ and BQ are equal to each other. Therefore, if AQ=6 then BQ=6 and AB=12. Angles in the Plane Some basic facts about angles formed in a plane by lines, line segments and rays. 1. Vertical angles and supplementary angles: two opposite angles formed by two intersecting lines are called vertical angles. 2. Vertical angles have the same measure. Angles next to each other add up to 180 degrees. Two angles that add up to 180 are called supplementary angles. Parallel lines: if a line intersects a pair of parallel lines, eight angles are formed. 4. Right angle: an angle with a measure of 90 degrees. 5. If two lines intersect and one of the four angles formed is a right angle, then the lines are perpendicular. 6. Two angles whose measures have a sum of 90 degrees are called complementary angles. 3. Triangles (Including Special Triangles) The sum of the measures of the angles in any triangle is 180. Equilateral Triangles The three sides of an equilateral triangle are equal in length. The three angles are also equal, so they each are 60 degrees. Isosceles Triangles An isosceles triangle is a triangle with two equal sides. The angles opposite the two equal sides are equal. Right Triangles and the Pythagorean Theorem A right triangle is a triangle with a right angle (90 degrees) So if one angle is 90 then the other two angles are complimentary (add up to 90 degrees) You can find the lengths of a right triangle sides by using the Pythagorean Theorem: a2+b2=c2 Where c is the hypotenuse. 30-60-90 Triangles 30-60-90 Triangles The guy who named 30-60-90 triangles didn’t have much of an imagination. These triangles have angles of 30,60, and 90. What’s so special about that? This: The side lengths of 30-60-90 triangles always follow a specific pattern. Suppose the short leg, opposite the 30° angle, has length x. Then the hypotenuse has length 2x, and the long leg, opposite the 60° angle, has length x√3. The sides of every 30-60-90 triangle will follow this ratio of 1:√3:2. 45-45-90 Triangles A 45-45-90 triangle is a triangle with two angles of 45° and one right angle. It’s sometimes called an isosceles right triangle, since it’s both isosceles and right. Like the 3060-90 triangle, the lengths of the sides of a 4545-90 triangle also follow a specific pattern. If the legs are of length x (the legs will always be equal), then the hypotenuse has length x√2. 3-4-5 Triangles Because right triangles obey the Pythagorean theorem, only a specific few have side lengths that are all integers. For example, a right triangle with legs of length 3 and 5 has a hypotenuse of length = 5.83. The few sets of three integers that do obey the Pythagorean theorem and can therefore be the lengths of the sides of a right triangle are called Pythagorean triples. Here are some common ones: {3, 4, 5} {5, 12, 13} {7, 24, 25} {8, 15, 17} In addition to these Pythagorean triples, you should also watch out for their multiples. For example, {6, 8, 10} is a Pythagorean triple, since it is a multiple of {3, 4, 5}. The SAT is full of right triangles whose side lengths are Pythagorean triples. Study the ones above and their multiples. Identifying Pythagorean triples will help you cut the amount of time you spend doing calculations. In fact, you may not have to do any calculations if you get these down cold. Congruent Triangles Are identical. Two triangles are congruent if they meet any of the following criteria: All corresponding sides are equal. This is known as the Side-Side-Side (SSS) method of determining congruency The corresponding sides are equal, and the mutual angles between those corresponding sides are also equal. This is known as the Side-Angle-Side (SAS) method of determining congruency The two triangles share two equal corresponding angles and also share any pair of corresponding sides. This is known as the Angle-Side-Angle (ASA) method of determining congruency Similar Triangles Corresponding angles of similar triangles are identical. Corresponding sides of similar triangles are proportional. The Triangle Inequality The length of any side of a triangle will always be less than the sum of the lengths of the other two sides and greater than the difference of the lengths of the other two sides. Quadrilaterals There are relationships among the angles and sides in parallelograms, rectangles, and squares. Parallelograms Opposite angles are equal Opposite sides are equal Rectangles All angles are right angles Squares The lengths of all the sides are equal. Areas and Perimeters Rectangles and Squares • Area=length x width • Area = s2 • The perimeter is the sum of all sides. Or (length + width) times 2. • The perimeter of a square can be found by multiplying 4 times the length of any side. Triangles ◦ Area = (1/2)bh, where b is the base and h is the height. Parallelograms ◦ The area is length x height Other Polygons A regular polygon is a polygon whose sides are all the same length and angles have the same measure. You can find the total number of degrees in the interior angles by dividing the polygon into triangles. Since one triangle has 180 degrees for its interior angles, just multiply the number of triangles times 180. Circles Diameter (D) = 2r Radius (r)= D/2 Length of Arc = D ∏(degrees of angle / 360) Tangent to a Circle Circumference= ∏D Area= ∏ r2 Solid Geometry (lots to remember) Volume of the cube = s3 Surface area of a cube = 6s2 Volume of rectangular solid = lwh Surface area of rectangular solid = 2(lw + wh + lh) Volume of prism = area of base × height Volume of a Cylinder = ∏r2h Volume of a Sphere = 4/3∏r3 Surface Area of a Sphere = 4∏r2 Volume of a Cone = 1/3∏r2h Volume of a Pyramid=1/3 x area of base x height Coordinate Geometry Slopes, Parallel Lines, Perpendicular Lines For a great explanation on coordinate geometry, go to http://www.math.com/school/subject3/less ons/S3U1L2DP.html . Midpoint Formula A midpoint is a point that denotes the middle of any given line segment. The Midpoint Theorem says the x coordinate of the midpoint is the average of the x coordinates of the endpoints and the y coordinate is the average of the y coordinates of the endpoints. If a line segment has the end points (x1, y2) and (x2, y2), the midpoint is given by the following formula: [((x1 + x2)/2), ((y1 + y2)/2)]. Distance Formula The distance formula says that the distance d between any two points with coordinates (x1, y1) and (x2, y2) is given by the following equation: d = SQRT[(x2 - x1)2 + (y2 - y1)2]. Problem: Find the distance between (-2, 3) and (8, -1). Solution: Plug any given information into the distance equation. d = SQRT[(8 - (-2))2 + (-1 - 3)2] Simplify. d = SQRT[102 + (-4)2] d = SQRT(100 + 16) d = 2(SQRT(29)) This information and more can be found at http://library.thinkquest.org/20991/geo/coordgeo.html Transformations A transformation is when you take a figure that is drawn on a plane and then move the figure. Three ways for a figure to ‘transform’ are: ◦ Rotation (turn) ◦ Reflection (flip) ◦ Translation (slide) Remember that the figure retains the same size and shape – you just adjust the points of the figure by moving up or down according to the problem. Chapter Nineteen Multiple-Choice Questions Recap on pages 333-334 of your book Approaches 1. 2. 3. Make sure you know what the question is asking and that you know what information the question gives you. Answer the question and check your answer by making sure your answer makes sense. Do your work in the test booklet. Draw out figures, highlight key information in the question, mark each question that you do not answer so you can go back to it if you have time and cross out answers that you know are incorrect. Approaches, cont. 4. 5. 6. Substitute numbers – if you substitute a number it makes the equation more concrete Substitute answer choices – take one of the answer choices and plug it into the equation. If you can eliminate at least one answer choice, make an educated guess. Approaches, cont. 7. 8. 9. If you have a question where a group of fractions is being multiplied, see if you can cancel out any of the numbers. When you are checking the values of expressions, remember the rules for multiplying positive and negative numbers. Look at each Roman numeral answer choice as a separate true-false question. Approaches, cont. Read word problems carefully. 11. Label diagrams and figures with the information you have. 12. If a figure is not given, draw the lines and figures that are described in a question. 13. Check that the picture you have drawn is consistent with the information given in the problem. 10. Approaches, cont. Find information about lengths and angles from your knowledge of the coordinate system. 15. Check that your answer observes the conditions of a problem. 16. If you have a special symbol in a problem, read the definition carefully. 17. Make sure that you follow the instructions carefully when you draw figures to help you solve logic problems. 14. Approaches, cont. 18. Rely on the accuracy of the grid lines shown in a graph.