Download ACT Overview - Thomas Herrington

ACT Review
Chapter One
Start Here
Overview of PowerPoint
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This PowerPoint will cover:
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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.
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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.
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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
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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.
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How is the ACT Scored..con’t
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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.
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How to Take It?
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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
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ACT Registration
P.O. Box 414
Iowa City, IA 52243
319-337-1270
ACT vs. SAT
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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.
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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!
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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.
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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.
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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.
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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.
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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.
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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.
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The End
Chapter Three
Guessing and POE
Table of Contents
No Penalty
 POE
 Letter of the Day
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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.
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POE
POE stands for Process of Elimination.
 If you do not understand a question, start
with the answers and work backwards.
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POE Example
Look at this question:
 1. What is the capital of Myanmar?
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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.
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Letter of the Day, cont.
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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
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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??
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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
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Don’t Leave Home Without It
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You will need these items on the test day:
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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
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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.
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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.
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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.
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Watch out… con’t
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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.
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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
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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.
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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
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Example Passages
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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.
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Look for Clues
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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.
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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.
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POE, No Change and OMIT
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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
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Basic Terms you need to know:
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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
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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
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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.
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Independent Clause. New independent
clause
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Example:
◦ John went to school. His mom went to work.
Independent Clause, (conjunction)
independent clause.
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John went to school, and his mom went
to work.
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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…
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A comma goes before the conjunction
that joins the two independent clauses.
Independent clause; independent
clause.
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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.
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Independent clause, dependent clause.
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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.
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Dependent clause, independent clause.
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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.
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Independent Clause
and a Modifying Phrase
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Tired and upset, she flopped down on the
couch and immediately went to sleep.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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Bad things, man. Really bad things.
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Do not, I repeat, do not make these
simple, careless errors:
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Sentence fragments
Run-ons
Comma splices
Misplaced modifiers
Non-parallel construction
Sentence Fragments
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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.
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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…
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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.
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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?
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Modifying Phrases
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If a sentence begins with a modifying
phrase – just like in the example – it
needs to be followed by the noun that it
modifies.
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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.
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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.
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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.’
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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
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You better know it: It’s, Its’, Its
It’s = It is
 Its’ = Is not a word
 Its = possessive of its
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References
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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.