Transcript x, -y


Chapter 7 - Transformations (10)
translations, reflections, rotations, line of
symmetry, rotational symmetry, vectors,
glide reflections
Reflections on the x axis, the y
axis & the line y = x
 Over x-axis
(x, y)  (x, -y)
Over y-axis
(x, y)  (-x, y)
Over the line y = x
(x, y)  (y, x)
 90º
counterclockwise, 270 clockwise
about the origin
(x, y)  (-y, x)
• If a point is rotated 90° clockwise, 270
counterclockwise about the origin, then
(x , y)  (y, -x)
 If
a point is rotated 180° about the
origin, then
(x, y)  (-x, -y)
For a translation:
(x,y)
 (x+a, y+b) or a,b
(x+a,
y+b) is the coordinate
notation
a,b> is the vector in
component form

Chapter 8 – Similarity (12)
definition of similarity, proportions,
similarity theorems, scale factor, dilations
Similarity: sides proportional
and angles congruent!!!
RATIO: A comparison of two or more
quantities in the same unit.
 written
: a/b or a:b.
Proportion: An equation which states two
ratios are equal.
a c

b d
ad  bc
a  bb c  d
 
cb d
d
a c

b d
a  bb c  d
 
cb d
d
a c

b d
Addition property of Proportions:
a c

b d
ab cd

b
d
a c

b d
a b

c d
Geometric Mean:
a x

x b
a  bb c  d
 
cb d
d
a c

b d
a  bb c  d
 
cb d
d
a c

b d
Angle-Angle Similarity postulate:
If two angles in one triangle are
congruent to two angles in another
triangle then the triangles are similar
SSS Similarity Theorem: If the corresponding
sides of two triangles are proportional, then the
triangles are similar.
SAS Similarity Theorem: If an angle in one
triangle is congruent to an angle of a second
triangle and the lengths of the sides including
these angles are proportional, then the
triangles are similar.
Triangle Proportionality Theorem
(Side - Splitter Theorem)
Q
T
TU || QS if and only if
RT RU

TQ US
R
U
S
Parallel Lines & Proportions
r
If r || s and s || t, and l and
m intersect at r, s, and t,
then UW  VX .
WY
XZ
s
t
l
U
W
Y
V
X
Z
m
Angle Bisector Theorem

If a ray bisects an angle of a triangle, it
divides the opposite side into
segments that are proportional to the
adjacent sides.
If CD bisects ACB ,
then AD  CA .
DB
CB
A
D
C
B
Dilation
SCALE FACTOR: Ratio of any length on the
image to the corresponding length on the
original figure.
image
preimage(original )
To create a dilation about the origin:
Multiply each coordinate by the scale factor

Chapter 9 – Right Triangles & Trig (13)
Pythagorean Theorem, Pythagorean
triples, classifying triangles, altitude on
hypotenuse theorem, geometric mean,
45-45-90 right triangle, 30-60-90 right
triangle, trigonometric ratios.
If the altitude is drawn to the hypotenuse of a right triangle,
then the two triangles formed are similar to the
original triangle and to each other.
ABC ~ CBD ~ ACD
The length of the altitude is the geometric
mean of the two segments.
AD CD

CD BD
The length of each leg is the geometric mean of the
lengths of the hypotenuse and the segment of the
hypotenuse that is adjacent to the leg
BA CA

CA AD
Pythagorean Theorem
Right Triangle: c 2 = a 2 + b 2
c
a
b
A Pythagorean triple is a set of three positive
integers a, b, and c that satisfy the equation
c 2 = a 2 + b 2.
Acute Triangle: c2 < a2 + b2
Obtuse Triangle: c 2 > a 2 + b 2
45 – 45 – 90
45
x√2
x
90
45
x
30 – 60 – 90
x
60
2x
90
x√3
30
Soh Cah Toa
opposite
Sin 
hypotenuse
adjacent
Cos 
hypotenuse
opposite
Tan 
adjacent
Solving Right Triangle
To find a side:
ouse the given angle and decide sin, cos, or tan
oset up an equation
oSolve for unknown variable
To find an angle:

use inverse sin, cos, or tan:


sinA = x, then sin-1(x) = A
cosA = x, then cos-1(x) = m
tanA = x, then tan-1(x) = m
Solving Right Triangle
Vector: Quantity with a magnitude and direction
Magnitude (speed): The distance from the initial
point to the terminal point
AB 
x2  x1    y2  y1 
2
2
Direction of a vector
Determined by the angle it makes with
a horizontal line
X axis: east-west
Y-axis: north-south

Chapter 10 – Circles (14)
definitions, radii-tangent theorems, measure of
angles with relations to circles, equation of a
circle
A set of all points equidistant from a given point
Circle: (called the center)
Radius: Segment from the center to a point on the circle
Diameter: Distance across the circle through the center

Chord: Segment whose endpoints are points on the circle

Secant: A line that intersects the circle in two points

Tangent: A line that intersects the circle at exactly one
point
More Circles
Internal Tangent: Tangent line intersects a line that joins
External
The centers of the circle
Tangent: Does not intersect the segment
that joins the center
Circle Theorems
Thm 10.1 If a line is tangent to a circle, then it is
perpendicular to the radius drawn to the
point of tangency
Thm 10.2 If a line is perpendicular to a radius of a circle
at its endpoint on the circle, then the line is
tangent to the circle.
Circle Theorems
Thm 10.3 If two segments from the same exterior point
are tangent to a circle, then they are congruent.
x
y
x and y are tangent to the circle, so x = y .
Angles and Arc Relationships

Central angle = arc


Inscribed angle = ½(arc)


Angle inside circle = ½(sum)

Angle outside circle = ½(difference)
Theorem

inscribed angles intercept the same arc, then
the angles are congruent
B
A
C

D
A  B
Since both angles intercept arc CD

Theorems:
A right triangle is inscribed in a circle  the
hypotenuse is a diameter of the circle.

A quadrilateral can be inscribed in a circle if
and only if its opposite angles are
supplementary.
w  y  180
o
o
x
0
x  z  180
o
o
0
w
y
z
Segment Relationships in Circles
Two chords: (part)(part) = (part) (part)
Two secants: (whole)(outside) = (whole)(outside)
Secant and tan: tan2 = (whole)(outside)
Standard Equation of a Circle
(x
-
2
h)
+ (y -
2
k)
=
2
r
(h, k) is the center
 r is the radius

Note: To find the radius, you may need to
use distance formulas
d
x2  x1 
2
 ( y2  y1 ) 2

Chapter 11 – Area of Polygons & Circles (12)
area formulas for regular polygons, equilateral
triangles, circles, arc length & area of sectors
Polygon interior angles sum:
180(n – 2)
Each polygon interior angle: 1  (n  2) 180 o
n
Polygon Exterior angles sum: 360o
Each polygon interior angle: 360o/n
Area of an Equilateral Triangle
1
2
A
3s
4
Area of a Regular Polygon:
1
A  ans
2
360
central angle 
n
Areas of similar polygons:
a
Sides:
b
Perimeter: a
b
2
Area:
a
2
b
Circumference:
The distance around a circle
Formula to find circumference:
c  2r
Arc Length:
The length of a portion of a circle
Formula to find arc length :
mAB
Arc length of AB 
 2 r
360
Area:
The amount of space inside a circle
Formula to find area:
A  r
2
Area of a Sector:
The area of a piece of the circle
(bounded by two radii)
Formula to find the area of a sector :
mAB
2
Area of AB 
 r
360
Geometric Probability
part
Area shaded region

whole
Area entire region

Chapter 12 – Surface Area & Volume (14)
all surface area and volume formulas for
prisms, cylinders, cones, pyramids, & spheres
Eulers Theorem: The number of faces (F),
vertices (V), and edges (E) of a polyhedron are
related by the formula:
F+V=E+2
To find edges: take ½ of the number of sides of
each face
Why? each side is shared by two polygons
Surface Area of a Right Prism
S = 2B +ph
Surface Area of a Cylinder
S  2r  2rh
2
Surface Area of a Pyramid
S = B + ½Pl
Surface Area of a Cone
S  r  rl
2
Volume of Cube
V
3
=s
Volume of Prisms
V = Bh
Volume of Cylinder
V=
2
Πr h
Volume of Pyramids
V = Bh
3
Volume of Cone
V = Πr2h
3
Surface Area of a Sphere
S  4r
2
Volume of Sphere
4 3
V  r
3
Theorem:
If two similar solids have a scale factor of
a:b, then the corresponding areas have a
ratio of a2:b2, and corresponding volumes
have a ratio of a3:b3.