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
ab cd
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 2r
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 2r 2rh
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 4r
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.