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Geometry
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A math & science tutorial center
Distance Formula
(X2 – X1)2 + (Y2 – Y1)2
d=
Pythagorean Formula
c = a2 + b2
c = hypothenus
Midpoint Formula
X2 + X1
2
Y 2 + Y1
=
2
X mp =
Ymp
30-60-90 triangle
60
1
2
30
Sum of internal Angles in polygon
3
Sum = (n – 2)180o
45-45-90 triangle
= 360o for any polygon
1
45
2
Number of diagonal in polygon
45
1
n(n-3)
diagonals =
2
Concave polygon
At least one internal angles > 180o
Scalene Triangle
All sides are different
All angles are different
1 2
3 4
5 6
7 8
Vertical Angles
<5 = <8
<1 = <4
<7 = <6
<2 = <3
Corresponding Angles
<1 = <5
<4 = <8
<2 = <6
<3 = <7
Alternate Interior Angles
<5 = <4
<6 = <3
Sum of External Angles in polygon
Convex polygon
All internal angles < 180o
Angles & Parallel Lines
Right Triangle
One 90o angles
Acute Triangle
All angles < 90o
Obtuse Triangle
One angles > 90o
Alternate Exterior Angles
<1 = <8
<7 = <2
Consecutive Interior Angle
<3 + <5 = 180o
<4 + <6 = 180o
Congruency Tests
SSS
SAS
ASA
AAS
HL
Similarity Tests
SSS
SAS
AA
Kite
- One pair congruent angle
- Long diag. bisect short diag.
- Adjacent sides congruent
- Diagonals perpendicular
Rhombus
-4 congruent sides
-Opposite angles congruent
-Diagonals perpend. bisector
B
D
Equilateral Triangle
All 3 sides are same
All 3 angles are same
A
Rectangle
- Opposite sides are parallel
- Opposite sides are congruent
- Diagonals bisect each other
- Diagonals are congruent
- All four angles = 90o
Square
- Opposite sides are parallel
- All sides are congruent
- All angles = 90o
- Diagonals are congruent
- Diagonals are perpendicular
- Diagonals bisect each other
- Diagonals are angle bisectors
Trapezoid
- Bases are parallel
- Median is leg bisector
- Adjacent top+bottom angles = 180o
Isoceles Trapezoid
- Bases are parallel
- Legs are congruent
- Diagonals are congruent
- Lower base angles are congruent
- Upper base angles are congruent
- Adjacent top+bottom angles = 180o
Median
- Bisect opposite side
Altitude
- Perpendicular to opposite side
Right Triangle Similarity
Isoceles Triangle
Two sides are same
Two angles are same
Parallelogram
- Opposite sides are parallel
- Opposite angles are congruent
- Diagonals bisect each other
- Sum of interior angles = 360o
- Sum of adjacent angles = 180o
QuadrilateralsFamily Tree
C
∆BAC ~ ∆CAD ~ ∆BCD
(CD)2 = (AD)(BD)
(BC)2 = (AB)(BD)
(AC)2 = (AB)(AD)
quadrilaterals
trapezoid
Convex quadrilateral
parallelogram
sq
rectangle
Midsegment bisector
x
2x
If mid-segment is bisector
Then larger base 2x smaller base
Centroid
- Intersection of medians
- Balance point or center of mass
-Circumcenter
- Intersection of perpendicular bisector
- circumscribed circle
Incenter
- Intersection of angle bisector
- Inscribed circle
Orthocenter
- Intersection of altitudes
- Larger segment = 2 x small segment
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Geometry
FREE step-by-step solutions
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A math & science tutorial center
Inscribed Angle
Square
P = 4s
A=sxs
s
s
50
P 25
L
Triangle
a
c
b
120
P
60
a
P=a+b+c
A= bxh
2
h
2
D
<P =
B
Outside arc
A
2
P = 3s
s
s
50
Central Angle
m<O = same as arc
50
C
P
h
C
(PA)2 =
b
b1
P
A
s1
If cords parallel,
then arcs are congruent
Inside Circle Angle
m<COD = m<AOB = vertical angle
70
O
B
Trapezoid
C
/
50
60
outside arc + outside arc
2
70 + 50
< AOB =
= 60
2
< AOB =
b2
d1
A=
(b1 + b2 ) h
2
Rhombus
d2
/
d1 x d 2
2
If cords congruent,
then equidistance
from center
P=2πr
A = π r2
r
Outside Circle Angle
P
100
20
Right Cylinder
r
outside arc - inside arc
<P=
2
100 − 20
<P =
= 40
2
Prisms
V = BH
P
Regular Polygon
P = (n) (s)
(s) (a) (n)
A=
2
Regular Pyramid
H
W
V= lxwxh
3
Sector of circle
r
θ
Arc = (
A =(
θ
360
θ
360
LA = all sides
SA = all sides + base + top
250
100
Spheres
75
a
LA = 2πrh
SA = 2πrh + 2πr2
V = πr2h
h
Outside Circle Angle
circle
L
If radius perpendicular
then cord is bisected
/
40
P = 4s
A=
s
s
If cords congruent
then arcs are congruent
/
D
s2 P = b1 + b2 + s1 + s2
h
(PB)(PC)
B
A
P = 2b + 2c
A=bxh
(PA)(PA) = (PB)(PC)
PA = PB
Parallelogram
c
(PB)(PA) = (PD)(PC)
B
3 s2
4
A=
s
P
D
b
Equilateral Triangle
(AP)(PD) = (BP)(PC)
P
A
O
c
B
A
Inscribed Angle
P=a+b+c
A= bxh
2
h
Outside arc
C
Rectangle
P = 2L + 2w
A=Lxw
w
<P =
Segment Proportionalities
) (2π r)
) (π r 2 )
SA = 4πr2
r
outside arc - inside arc
<P=
2
250 − 100
<P =
= 75
2
V=
4πr3
3
Pyramids
H
Outside Circle Angle
LA = all sides
SA = all sides + base
V=
P
80
200
Cones
LA = π r s
60
s
outside arc - inside arc
<P=
2
200 − 80
<P =
= 60
2
ΒΗ
3
r
s
r
SA = π r s + π r 2
V=
π r2 H
3