Download General Formulas Polygons

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Geometry B
—
Semester Overview
General Formulas
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Distance Formula: Given P=(xi,yi) and Q
•
Slope Formula: Given P=(xi,yj) and Q
•
=
=
(x2,y2),
distance is d
(x2,y2), slope is m
=
J(x —x
1
+(y 2
2 1
)
7
—y,)
=
xl —x
2
Midpoint Formula: Given P=(xj,yi) and
2
±X
Q (x2,y2), midpoint is
=
Y ±Y
2
Polygons
Poly2on Characteristics:
•
•
•
•
•
•
•
•
Closed flat geometric figures
No side is curved
Consecutive sides intersect oniy at endpoints
Nonconsecutive sides don’t intersect
Each vertex must belong to exactly 2 sides
Consecutive sides must be non-collinear
Each segment of polygon is called side and each endpoint of side is called vertex.
The number of sides is always equal to the number of vertices.
Identifying Po1yons:
•
•
Name each polygon by the number of its sides.
Identify if convex/concave and regular/irregular.
o Regular convex polygon where all sides/angles are congruent
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Poiygon Formuias:
•
Sum Si of measures of angles of polygon with n sides: S
•
If I exterior angle is taken at each vertex, sum S of measures of exterior angles of polygon: Se
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Number d of diagonals that can b drawn in polygon of n sides: d
•
Measure E of each exterior angle of equiangular polygon of n sides: E
•
Sum of measure of 3 angles of triangle is 180
•
Exterior Angle of Polygon: adjacent & supplementary to interior angle of polygon. Measure equals
sum of measures of remote interior angles.
=
(n-2)180
=
360
n(n —3)
RevA
Geometry B
—
Semester Overview
Proportions/Similarity
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•
•
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Perimeter Ratio Theorem: The ratio of the perimeters of 2 similar polygons equals the ratio of any
pair of corresponding sides.
AA’-. Theorem: if 2 Ls of I A are to 2 /s of other A. then As are
SSS Theorem: if measures of corresponding sides of 2 As are proportional then As are
SAS- Theorem: if measures of 2 sides of A are proportional to measures o12 corresponding sides of
then As are
another A and included angles are
Given the diagram to the right, the following apply
A
Bz’ E
ifBEJCD.=-(B ED
C”z’
D
ir-=
then BE Ci)
ED
(B
if B and E are midpoints of AC and
•
CD and BE
CD
=
Three Parallel Lines Corollary: if 3 or more parallel lines are intersected by 2 transversals, the
A
parallel lines divide the transversals proportionally.
If AB
H CD H
EF,
BE
(E 4E
BD AE
4(
=
=
DF
DF CE
DE BE
CE
angle
of triangle. it divides opposite side into segments
an
Angle Bisector Theorem: if a ray bisects
that are proportional to adjacent sides.
then
•
then BE
—
—.
—
——,
If AC bisects ZBAD then
BC[)
•
Proportions: 2 or more ratios are equal
o a is l term
a = c
uid
or a:b = c:d
term
a b is 2
rd
term
oc is 3
th
term
ad is 4
-—
—
th
4
terms
o Extremes: 1 &
terms
a Means: 2 &
•
Mean ProportionallGeometric Means: (means are the same i.e. aix
o Example: find geometric means of I and 16
• 1/x = x/l6 (Set the proportions)
• I * 16 = xx (Cross multiply)
16 = x (Simplify)
• \u16x2 (Solvetbrx)
• +4x
•
Given 2 squares with sides x and y. ratios of
o Perimeters is x:v
2
o Area is x:y
2
‘
=
x/r)
RevA
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____________
Geometry B
—
Semester Overview
Right Triangles/Trigonometry
Finding missing lengths of right triangle
When an altitude is drawn from the vertex of a right angle of the right triangle to its hypotenuse. then the
Ibllowing apply:
A
• Altitude geometric means between
c
1
measures of the 2 segments of the hypotenuse
b
o
alt=
2
,Jci*c
C2
•
Sides geometric means of hypotenuse and
part of hypotenuse adjacent to the side.
o a+c,)
o
C
a
B
hc(c+c,)
Classify Type of triangle: c is length of longest side of triangle
o
o
o
a
2
a
2
a
±
+
+
h>c
2
2 triangle is acute
2 triangle is right
2=c
b
<c triangle is obtuse
22
b
2
Pythagorean Theorem: a
o
+ h
2=c
2
Triples: 3-4-5. 5-12-13, 7-24-25, 8-15-17. 9-40-41
Special Right Triangles
l
Solving Right Triangles Trig Functions
-
•
SOl I-CAB-bA
•
Sin
=
•
•
=
LI
A
CosA
Tan A
h’poic’nuse
c
ad/uceni
h
hvpownuse
c
opposite
adjacent
a
=
C
h
3
b
A
RevA
Geometry B
—
Semester Overview
Circles
Formulas
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•
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Radius = /2 Diameter
Circumference of Circle: dir or 22tr
2
Area of Circle = irr
2
2 + (y-k)
2=r
Equation of a circle: (x-h)
Key Concepts
•
•
•
•
.
All radii are congruent
Vertex on Center: angle = intercepted arc
2 (intercepted arc)
Vertex on Circle: angle = V
2 (arc 1 + arc2) = average of bases
Vertex inside circle: angle = V
Vertex outside circle: angle = V2 (arc 1 arc2) difference of arcs
—
Finding the angle measure:
Tangent-Secant
Chord-Chord
Tangent-Tangent
Secant-Secant
Inscribed Angle
c
(c—b)
2
a= V
Finding the segment measure:
Chord-Chord
a
=
½
a
c
rr
½
Secant-Secant
Tangent-Secant
14
x
I6
32
32(x)
16(12)
I
2
142
=
.1:
7(7 + x)
4
/1
AB(AC)
=
AD(AF)
RevA
(c
-h)
Geometry B
—
Semester Overview
Area
•
If 2 closed regions intersect only along a common boundary, then the area of their union is equal to the
sum of their individual areas. (This means break it up into pieces, find area of each piece and then add
areas together.)
Formulas:
Formula
Notes
r
Circle
Kite
1
A
dd
212
Parallelogram
A
=
bh
Rectangle
A
=
1w or A
Rhombus
Square
Trapezoid
is the radius
d’s are the lengths of the diagonals of the kite
A
h is the perpendicular height to the base
=
bh
or A
2
Length x width or
-
base x height
bh
2
As
A
h is the altitude
1
=
—h(b +b
1
)
2
2
and
the bs are the bases.
M is the median of the Trapezoid
Or
A=Mh
Triangle
A
Equilateral Triangle
=
—
s is the side of the triangle
s2T5
A=
TriangleHero’s
h is the altitude to the base
!hh
2
4
AJ—s7)?c)
Formula
(a+b+c)
2
Use when there is no right angle or altitude
Regular Polygon
Sector of Circle
A
I
P is the perimeter of polygon
a
&
a is the
apothem
N
A=—izr
360
N is the degree of the arc
,
5
RevA
Geometry B
—
Semester Overview
Surface Area & Volume
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Volume: area of base x height
Lateral Surface Area: sum of areas of lateral faces
Surface Area: lateral surface area ± area of bases
Triangular Prism: remember the base is the triangle not the rectangle.
Note: The following apply to the k)rmulas below:
o Ph Perimeter of the base
o
Ab Area of the base
/)
o B = area of the base
a
= slant height
Sphere
•
•
•
=
4iti
•
4
Vsphere =
3
Radii of 2 spheres with ratio x:y
o
Ratio of volumes: 3
:y
x
o Ratio of surface areas: 2
:y
x
=
•
SAcone
011
LA
=
Bh
Vtiem,spiiere
+
Aconchase
itr (‘
=
r
A
Lzpyra
rnij
+
2
mr
Note: use slant height for surface area and
height for volume,
SArcet
=
2(lh)
+
Vreet
=
•
=
LArect
—
2
•
SApyrarnid
•
Vpyraniid
•
Note: use slant height for surface area and
height for volume.
=
LApyranhid
+
Apr,c
Ahh
Cube
=
+
2(hw) +
Bh lwh
2(l+w)
*
h
2*A hasc 2(l+w)h
2(1w)
+
2(1w) or
Triangular Prism
•
•
•
=
3
=
Rectangle
• LArect = Ph * h
• A base = l*w
•
4 (2tr 1)
=
Aeoiiebase
•
2
37tr
2
SAheniispherc
Pyramid
Ch
=
•
•
C = circumference of the circle
h = height
hemisphere
SAsphere
Cone
• LA
11
•
a
a
•
•
•
LAb
SAb
=
4(s
)
2
6(s)
Bh lwh
=
=
3
s
Cylinder
LAtriaiig Pi, * h
A(riangbase = see area section for options
SAtriang LAiriang + 2
Atrianghasc
Viriajig = Bh
•
•
•
22tr
•
6
LAcylinder = Ch = 2itrh
Acslinderhase = itr
SAcylinder LAcyiincer ± 2
Acylindcrbasc
Veylinder
=
Bh
=
RevA
2rrh
+