Transcript Slides with Notes

Distance Formulas:
•
•
Number Line: b-a for a and b the
coordinates of the points
Plane:
( x2  x1 ) 2  ( y2  y1 ) 2
Exercises:
1.
2.
Find the distance AB if a = -3 and b=7
Find the distance MN if M(2,-1) and N(-2, 4)
Slopes
If m > 0, line rises from left to right
If m < 0, line falls from left to right
If m = 0, horizontal line
If no slope (undefined), vertical line
y2  y1
m
x2  x1
y  mx  b
Exercises:
1. Find the slope of the line that passes
through points (2,-5) and (0, -3)
2. Find the slope of the line perpendicular to
y=¾x–6
Midpoint:
 x1  x2 y1  y2 
,


2 
 2
Exercises:
1.
Find the midpoint of a segment whose endpoints
have coordinates (6, 7) and (-5, -5).
2.
If the midpoint of a segment is (2, -4), and one of
the endpoints is (0, 8), find the other endpoint.
Pairs of Angles:
 Complementary and supplementary angles
 Linear pair
 Vertical angles
Exercises:
Complete each statement.
1. If two angles are vertical angles, then they are
______________.
2. If two angles are a linear pair of angles, then they are
______________.
3. If two angles are both equal in measure and
________________, then each angle measures 90°.
Parallel Lines
Congruent
Alternate Interior
Alternate Exterior
Corresponding
Vertical
Supplementary
Consecutive Interior
Unrelated
Linear Pair
More Parallel Lines
Exercises
Find the measure of each angle indicated in the figure below.
r
62° f
d
g
h
c
j
b
a
s
28°
e
t
w
Quadrilaterals:
Quadrilaterals
Parallelograms
Rectangle
Kites
Rhombus
Square
Trapezoids
Isosceles Trapezoids
Parallelograms
• Opposite sides are parallel
• Opposite sides are congruent
• Opposite angles are congruent
• Consecutive angles are
supplementary
• Diagonals bisect each other
J
K
L
M
Rectangles
O
• Opposite sides are parallel
• Opposite sides are congruent
N
• Opposite angles are congruent
• Consecutive angles are supplementary
• Diagonals bisect each other
• Right angles
• Congruent diagonals
P
Q
Rhombus
F
• Opposite sides are parallel
• Opposite sides are congruent
E
• Opposite angles are congruent
• Consecutive angles are supplementary
• Diagonals bisect each other
• Congruent sides
• Perpendicular diagonals
• Diagonals bisect both pairs of opposite
angles
G
H
Squares
• Opposite sides are parallel
• Opposite sides are congruent
• Opposite angles are congruent
• Consecutive angles are
supplementary
• Diagonals bisect each other
• Right angles
• Congruent diagonals
• Congruent sides
• Perpendicular diagonals
• Diagonals bisect both pairs of
opposite angles
C
A
D
B
Trapezoids
 Exactly one pair of opposite sides
are parallel — bases
 Non parallel sides are the legs
 Median: half of the sum of the
bases
 Isosceles Trapezoid:
 Congruent legs
 Base angles are congruent
 Congruent diagonals
S
R
T
U
Kites
X
o No parallel sides
o Consecutive sides are congruent
o Diagonals are perpendicular
o Non vertex angles are congruent
o One diagonal splits the kite in congruent
halves
W
Y
V
More quadrilaterals:
Match each statement:
a. Rhombus
b. Rectangle
c. Trapezoid
d. Parallelogram
1.
2.
3.
4.
5.
i. Parallel
j. Supplementary
k. Complementary
_______ Two angles whose measures add up to 180°
_______ Two lines in the same plane that do not intersect
_______ An equiangular parallelogram
_______ A quadrilateral with exactly one pair of parallel sides
_______ An equilateral quadrilateral
More Exercises
Fill in the blanks
1. ___?____ An equiangular parallelogram
2. ___?____ A quadrilateral with exactly one
pair of parallel sides
3. ___?____ An equilateral quadrilateral
True or False
4. _____ A trapezoid is a quadrilateral having
exactly one pair of equal length sides.
5. _____ A parallelogram is a quadrilateral
with all the angles equal in measure.
Triangles
Triangles
Angles
Acute
Obtuse
Sides
Right
Equilateral
Isosceles
Scalene
Congruent vs. Similar
Congruent
Similar
 Same shape
 Same size
 Congruent angles
 Congruent sides
 Shortcuts:
 Same shape
 Different size
 Congruent angles
 Proportional sides
 Shortcuts:
 SSS
 SAS
AAS or ASA
 SSS
 SAS
 AA
Right Triangles
Pythagorean Theorem (and its converse):
a b c
2
2
a
2
b
Special right triangles:
•30°-60°90°
• 45°-45°-90°
2
c
1
2
1
1
3
Exercises
1. What is the length of the hypotenuse of a right triangle with
legs that measure 80 feet and 150 feet?
2. What is the length of the larger leg of a 30-60 right triangle
with a hypotenuse of length 24 m?
3. If the area of a square is 225 cm2, what is the length of the
diagonal?
4. In an isosceles right triangle, if the hypotenuse has length
x then each leg has length —?—.
5. In a 30-60 right triangle, if the hypotenuse has length y,
then the shorter leg has length —?— and the longer leg has
length —?—.
Trigonometric Ratios
opposite
 sin 
hypotenuse
adjacent
hypotenuse

cos ine 

opposite
tan gent 
adjacent
Exercises using trig ratios:
Calculate each distance or angle:
1.The angle of elevation from a ship to the top of a 40 meter lighthouse on the
shore is 18°. To the nearest meter, how far is the ship from the shore?
2.Igor is flying a kite with 300 m of kite string out. His kite string makes an angle
of 64° with the level ground. To the nearest meter, how high is his kite?
Use a calculator to find the values accurate to four decimal places:
1. sin 57° » –?–
2.cos 9° » –?–
Find the measure of each angle to the nearest degree:
3. sin A = 0.5447
4.cos B = 0.0696
Polygons Facts:
 Exterior angle sum: 360°
 Interior angle sum: 180°(n – 2)
 Exterior and interior angles are
supplementary
 Concave vs. convex
 Classification according to the
number of sides
Exercises
Complete each statement:
1. The sum of the measures of the interior angles of an
15-gon is ________.
2. The sum of the measures of the exterior angles of a 25-gon is
__________________.
3. The measure of one interior angle in a regular octagon is
_____________.
4. If the measure of one exterior angle of a regular polygon is 24°,
then the polygon has ______ sides.
Independent VS. Dependent Events
• Two events are said to be independent if the result of
the second event is not affected by the result of the first
event.
• If A and B are independent events, the probability of
both events occurring is the product of the probabilities
of the individual events
Mutually Exclusive/Inclusive Events
The Counting Principle
Examples
Permutations and Combinations
Systems of Linear Equations
The solution of a system of linear
equations in two variables is any ordered
pair that solves both of the linear
equations. The solution to the system is
the point that satisfies ALL of the
equations. This point will be an ordered
pair.
IDENTIFYING THE NUMBER OF SOLUTIONS
NUMBER OF SOLUTIONS OF A LINEAR SYSTEM
y
y
y
x
x
x
Lines intersect
Lines are parallel
Lines coincide
one solution
no solution
infinitely many solutions
Transformations
• Image – is the new figure. Image after the
transformations.
• Pre-image – is the original figure. Image
before the transformations.
• Transformation – In a plane, a mapping for
which each point has exactly one image
point and each image point has exactly one
pre-image point.
To transform something is to change it. In geometry, there are
specific ways to describe how a figure is changed. The
transformations you will learn include:
•Translation
•Rotation
•Reflection
•Dilation
Given (x, y) is the pre-image, then…
Parabolas
Vertex Form
y = a(x-h)2+k
Standard Form
y = ax2+bx+c
Exponential Functions
An exponential function is a function
of the form y  a  b ,
where a  0, b  0, and b  1,
and the exponent must be a variable.
x
Examples of Exponential Graphs
When b > 1 the graph increases and decreases when 0 < b < 1
y=2
x
æ1ö
y =ç ÷
è2ø
x
The Equality Property for Exponential Functions
Suppose b is a positive number other
than 1. Then b = b
x1
x2
if and only if
x1 = x 2 .
Basically, this states that if the bases are the same, then we
can simply set the exponents equal.
This property is quite useful when we
are trying to solve equations
involving exponential functions.
FINDING INVERSES OF LINEAR FUNCTIONS
An inverse relation maps the output values back to their
original input values. This means that the domain of the
inverse relation is the range of the original relation and
that the range of the inverse relation is the domain of the
original relation.
Original relation
x
y
– 2 DOMAIN
–1 0 1
4
Inverse relation
2
x
RANGE
2 0 –2 –4
y
4 DOMAIN
2 0 –2 –4
– 2 RANGE
–1 0
1
2
Logarithms
Base
Index
Power
Exponent
Logarithm
2
10
= 100
Number
“10 raised to the power 2 gives 100”
“The power to which the base 10 must be raised to give 100 is 2”
“The logarithm to the base 10 of 100 is 2”
Log10100 = 2
Logarithms
y = bx
Logby = x
Logarithm
Base
102 = 100
Number
logby = x
is the inverse of
y = bx
Base
Logarithm
Log10100 = 2
Number
23 = 8
Log28 = 3
34 = 81
Log381 = 4
Log525 =2
52 = 25
Log93 = 1/2
91/2 = 3
Graphs of Trig Functions