Upload
marshall-martin
View
229
Download
6
Tags:
Embed Size (px)
Citation preview
Read each question carefully.
Read the directions
for the test carefully.
For Multiple Choice Tests•Check each answer – if impossible or silly cross it out.
•Back plug (substitute) – one of them has to be the answer
•For factoring – Work the problem backwards
•Sketch a picture
•Graph the points
•Use the y= function on calculator to match graphs
Do the Easy Ones First Then go Back and do the Hard Ones!
Beware of the Sucker Answer
Make sure you answer the question that is asked!
Double check the question before you fill in the bubble!!
X 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
3 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45
4 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60
5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
6 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90
7 0 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105
8 0 8 16 24 32 40 48 56 64 72 80 88 96 104 112120
9 0 9 18 27 36 45 54 63 72 81 90 99 108117 126135
10 0 10 20 30 40 50 60 70 80 90 100 110 120130 140150
11 0 11 22 33 44 55 66 77 88 99 110121 132143 154 165
12 0 12 24 36 48 60 72 84 96 108120 132144 156168 180
13 0 13 26 39 52 65 78 91 104 117130 143156 169182 195
14 0 14 28 42 56 70 84 98 112 126140 154168 182196 210
15 0 15 30 45 60 75 90 105 120 135150 165180 195 210225
Factors Multiples Perfect Squares(6 ) (4 ) = 24
Geometry Basics
Point (Name with 1 capital letter)A
AB�������������� � • •
A BLine (Name with 2 capital letters, )
AB��������������
• •A B
Ray (Name with 2 capital letter, )
ABCAngle (Name with 3 letters. Middle letter is vertex
)B
A
C
•A B
Line Segment (Name with two letters, AB)
Plane (Name with 3 non-collinear points, ABC)A B
C
90 Complementary Angles Right Angles Symbol (┌ or ┐) Perpendicular ┴ A corner
180 Straight Angle (line) Supplementary Angles Half Circle Sum of angles in a triangle
360 Circle Sum of angles in a 4 sided figure (quadrilateral)
Also called linear pair
Complementary AnglesRight AnglesSymbol (┌ or ┐)Perpendicular ┴A corner
90
Straight Angle (line)Supplementary AnglesLinear PairHalf CircleSum of angles in a triangle
180
These two angles (140° and 40°) are
Supplementary Angles, because they add up
to 180°.
Notice that they are also a linear pair.
But the angles don't have to be together.
These two are supplementary because 60° +
120° = 180°
Supplementary Angles
Two Angles are Supplementary if they add up to 180 degrees.
Thanks to http://www.mathsisfun.com/geometry/complementary-angles.html
HINT: S Straight or S Splits
In this example, a° and b° are vertical angles.
a° = b°
Vertical Angles
Angles opposite each other when two lines crossThey are called "Vertical" because they share the same Vertex (or corner point)
http://www.mathwarehouse.com/geometry/angle/interactive-vertical-angles.php
Vertical angles are congruent and their measures are equal:
vertex
These two angles (40° and 50°) are
Complementary Angles, because they add
up to 90°.
Notice that together they make a right angle.
But the angles don't have to be together.
These two are complementary because 27° +
63° = 90°
Complementary Angles
Two Angles are Complementary if they add up to 90 degrees (a Right Angle).
Thanks to http://www.mathsisfun.com/geometry/complementary-angles.html
HINT: C Corner or C looks like a corner
Linear Pairs
Angles on one side of a straight line will always add to 180 degrees.
If a line is split into 2 and you know one angle you can always find the other one by subtracting from 180
25°A°
A° = 180 – 25°A° = 155°
Right AnglesA right angle is equal to 90°
Notice the special symbol like a box in the angle. If you see this, it is a right angle. 90˚ is rarely written. If you see the box in the corner, you are being told it is a right angle.
90°90°
Notice: Two right angles make a straight line
Properties of Equality
• Addition Property: If a=b, then a+c=b+c
• Subtraction Property: If a=b, then a-c=b-c
• Multiplication Property: If a=b, then ac=bc
• Division Property: if a=b and c doesn’t equal 0, then a/c=b/c
• Substitution Property: If a=b, you may replace a with b in any equation containing a and the resulting equation will still be true.
Properties of Equality
Reflexive Property:
For any real number a, a=a
Symmetric Property:
For all real numbers a and b, if a=b, then b=a
Transitive Property:
For all real numbers a, b, and c, if
a=b b=c a=c a=c
Conditionals & Bi-conditionals
EXAMPLES:
IF today is Saturday, THEN we have no school.
“IF-THEN ” statements like the ones above are called CONDITIONALS.
To make a bi-conditional, take off the IF and replace the THEN with “IF AND ONLY IF”
Today is Saturday, IF AND ONLY IF we have no school.
Conditional statements have two parts…
The part following the word IF is the HYPOTHESIS
The part following the word THEN is the CONCLUSION
IF today is Saturday, THEN we have no school.
Hypothesis: today is Saturday
Conclusion: we have no school
Conditionals
ConverseThe of a conditional statement is formed by
exchanging the HYPTHESIS and the CONCLUSION.
CONDITIONAL: IF it is snowing, THEN we will have a snow day.
IF we will have a snow day, THEN it is snowing.
CONVERSE
CONVERSE:
CounterexampleA Counterexample is an example that proves a statement false.
Conditional Statement: IF an animal lives in water, THEN it is a fish.
* This conditional statement would be false. You can show that the statement is false because you can
give one counterexample. *
Counterexample: Whales live in water, but whales are mammals, not fish.
If-Then Transitive PropertyGiven
If A then B, and if B then C.
You can conclude: If A then C.
If sirens shriek,
then dogs howl
If dogs howl,
then cats freak.
If sirens shriek,
then cats freak.
Quadrilaterals
Parallelogram
Rectangle
Rhombus
Square
Kite
Trapezoid
Isosceles Trapezoid
( 4 sides )
Rectangle
Congruent SidesCongruent AnglesParallel Sides
All angles are congruent (90 ˚ )
Diagonals are congruent
Parallelogram
Congruent SidesCongruent AnglesParallel Sides
Opposite sides
Opposite angles
Consecutive angles are supplementary
Opposite sides parallel
Diagonals bisect each other
Rhombus
Congruent SidesCongruent AnglesParallel Sides
All sides are congruent
Diagonals are perpendicular
Diagonals bisect angles
Square
Congruent SidesCongruent AnglesParallel Sides
Diagonals are perpendicular and congruent
Diagonals bisect each other
All sides are congruent
All angles are congruent
Angles = 90°
Isosceles Trapezoid Diagonals are congruent
Trapezoid Congruent SidesCongruent AnglesParallel Sides
Kite Congruent SidesCongruent AnglesParallel Sides
Diagonals are perpendicular
Geometry in Motion
ReflectionReflection
Transformation
Dilation
Rotation
Refl
ectio
n
Refl
ecti
on
A translation "slides" an object a fixed distance in a given direction. The original object and its translation have the same shape and size, and they face in the same direction.
Translations
Move up/downMove right/left
Let's examine some translations related to coordinate geometry.
In the example, notice how each vertex moves the same
distance in the same direction.
6 units to the right
TranslationsIn this next example, the "slide" moves the figure7 units to the left and 3 units down.
There are 3 different ways to describe a translation
1. description: 7 units to the left and 3 units down.
2. mapping: (This is read: "the x and y coordinates will become x-7 and y-3". Notice that movement left and down is negative, while movement right and up is positive - just as it is on coordinate axes.)
3. symbol: (The -7 tells you to subtract 7 from all of your x-coordinates, while the -3 tells you to subtract 3 from all of your y-coordinates.)This may also be seen as T-7,-3(x,y) = (x-7,y-3).
Reflecting over the y-axis:
When you reflect a point across the y-axis, the y-coordinate remains the same, the x-coordinate changes!
The reflection of the point (x,y) across the y-axis is the point (-x,y).
Reflecting over the x-axis:
When you reflect a point across the x-axis, the x-coordinate remains the same, and the y-coordinate changes!
The reflection of the point (x,y) across the x-axis is the point (x,-y).
Examples of the Most Common Rotations
Counterclockwise rotation by 180° about the origin:
A is rotated to its image A'. The general rule for a rotation by 180° about the origin is
(x,y) (-x, -y)
Examples of the Most Common Rotations
Counter clockwise rotation by 90° about the origin:
A is rotated 90° to its image A'. The general rule for a rotation by 90° about the origin is
(x,y) (-y, x)
Dilations always involve a change in size.
Dilations
Dilations
Dilations
Dilations
Dilations
DilationsDilations
DilationsDilations
Dilations
You are probably familiar with the word "dilate" as it relates to the eye. The pupil of the eye dilates (gets larger or smaller) depending upon the amount of light striking the eye.
Dilation is the same shape as the original, but is a different size. The description of dilation includes
the scale factor and the center of the dilation. A dilation of scalar factor k whose center of
dilation is the originmay be written: Dk(x,y) = (kx,ky).
.
Dilations - Example 1: If the scale factor is greater than 1, the image is an
enlargement (bigger).
PROBLEM: Draw the dilation image of triangle ABC with scale
factor of 2.
OBSERVE: Notice how EVERY coordinate of the original triangle has been multiplied by the scale
factor (2).
HINT: Dilations involve multiplication!
Dilations Example 2: If the scale factor is between 0 and 1, the image is a reduction (smaller).
PROBLEM: Draw the dilation image of pentagon ABCDE with a scale factor of 1/3.
OBSERVE: Notice how EVERY coordinate of the original pentagon has been multiplied by the scale factor (1/3).
HINT: Multiplying by 1/3 is the same as dividing by 3!
5 6
7 8
1 2
3 4
Interior
ExteriorParallel Lines Angles
Linear Pairs Supplemental Add up to 180 ‹1,‹2 ‹3,‹4 ‹2,‹4 ‹1,‹3 ‹5,‹6 ‹7,‹8 ‹5,‹7 ‹8,‹6
Vertical Angles Congruent ‹1,‹4 ‹3,‹2 ‹5,‹8 ‹6,‹7
Alternate Interior Angles Congruent ‹3,‹6 ‹4,‹5
Alternate Exterior Angles Congruent ‹1,‹8 ‹2,‹7
Corresponding Angles Congruent ‹1,‹5 ‹3,‹7 ‹2,‹6 ‹4,‹8
Same Side Interior Supplemental add up to 180 ‹3,‹5 ‹4,‹6
Transversal
lines
Same Slope
Slopes are Negative Reciprocal
Flip and Change Sign
PARALLEL LINES
y2 – y1
x2 – x1
or
y = mx + b
slope
Slope – Intercept Form
y = mx + b
Slope- directions
RiseRun
Y Interceptwhere to start
It’s a line address
If the slope is a whole number, put it on a stick m = 2 slope is 2/1
To Graph:
y = 2X + 1 Starts at 1
Rise/run = 2/1
Directions are up 2, over 1
Example 1 Example 2y = -3X+ 0
y = -3X
Starts at 0
rise/run = 3/-1
Directions are up 3, over -1
Thanks to http://www.mathsisfun.com/equation_of_line.html
Example: Solve for Y2x – 7y = 12
Just 3 easy steps1. -7y = 12 – 2x X is offside, Penalty change signs2. -7y = (12-2x) Huddle up ( )3. y = (12-2x) / -7 Man on man defense
WATCH YOUR SIGNS!!
Linear Equations, Standard Form ax + by = cSolving for y, It’s a football Game
Y VS Everybody Else
Follow football rules
Play FootballLetters vs Numbers
Now you are ready to enter it into the calculator or graph it
Find Equation of the Line: y = mx + b
I need slope (m) & the y-intercept (b)
MY ANSWER:
y = x +
To find m – Solve the equation for y and use mor use the y2 – y1
x2 – x1 formula
To find b - Plug x, y and m into the line equation and solve for b.
Slope: m =
Midpoint: (x, y) = ( , )
Distance: d =
Sum of the interior measures:
Sum of the exterior measures: 360°
Measure of the interior angle in a regular polygon:
Measure of the exterior angle in a regular polygon: 360°
Formulas
Line Stuff
Polygons:
Figure # of Sides
# of Triangles
Sum of Interior Angles
(# of Triangles)(180)
Triangle 3 1 180 1 * 180Rectangle 4 2 360 2 * 180Pentagon 5 3 540 3 * 180Hexagon 6 4 720 4 * 180Octagon 8 6 1080 6 * 180n -gon n n-2 180 (n-2)
Sum of the Angles of a Polygon.
Sum of Exterior Angles is 360
Tiles or floors
Floor Plan
Floor Rugs
Acres
Examples of things you’d find the area
of.
Perimeter – Path around the Outside
No Trespassing – Go all the way Around!
Area of Plane Shapes
Triangle
A = ½b×h Square
Area = a2
RectangleArea = b×h
ParallelogramArea = b×h
Trapezoid (US)Area = ½(b1+b2)h
Circle
Area = πr2
Area Formulas
b
h
b
h
b1
h
a
b
h
r
b2
Area of Plane Shapes
Triangle
P = a + b + c
SquareP = 4a
RectangleP = 2b + 2h
ParallelogramP = 2b + 2a
Trapezoid (US)P = a + b1 + b2 + d
Circle
Circumference=2πrr = radius
Perimeter Formulas
a b
c
h
b
a d
b1
b2
a
b
a
r
a
bc
A
B C
Law of Sinessin(A) = sin(B) = sin(C) a b c
Law of Cosinesa² = b² + c² – 2bc * cos(A)
b² = a² + c² - 2ac * cos(B)
c² = a² + b² – 2ab * cos(C)
cos(A) = (a² – b² – c²) (-2ab)
To convert from:Degrees to radians – multiply by π
180
Radians to degrees – multiply by 180 π
Trigonometry for Any Triangle
3 Trig Functions:
SOH
ΘSin =hyp
opp
adj
TOA
ΘTan =oppΘ
CAH
Cos =hyp
adj
Trigonometry Functions(Be sure your calculator is in degrees)
Hyp is always across from right angle. Adj and Opp change depending on Θ
Trigonometry is the study of how the sides and angles of a right triangle are related to each other.
3 Sides:
1. Hypotenuse - Across from right angle. 2. Opposite - Across from angle Θ. 3. Adjacent – Next to angle.
Θ
Θadj
adjhyp
hyp
opp
opp
•Use chart to organize information•Set up ratio•Cross multiply•Solve for X
To Solve: Ex 1 Ex 2
Hyp 31
Adj x
Opp 10 23
Θ 41 Θ
Trig Func. tan Sin-1