ACT - Geometry - Study Guide
SIMILARITY
1.
F
2. Basic Proportionality Theorem
C
A
A
AB BC AC
If ∆ ABC ~ ∆ DEF then


DE EF DF
B
Angle Bisector Theorem
a c

c
a
b d
b
5.
4.
AB AC

BD CE
C
D
A  D
B  E
C  F
3.
BC DE then
E
D
B
If
E
Multi parallel line, a // b // c
x
y
d
a
z
b
f
c
Ratio of perimeters = ratio corresponding sides
b
B
c
D
If ∆ ABC ~ ∆ DEF
d
e
a
A
perm ABC c
a
b
 or or
perm DEF f
d
e
F
C
E
f
RIGHT TRIANGLE
1. Altitude on the hypotenuse theorems Redraw the triangles and use similarity
B
C
D
A
D
D
A
B
B
B
C
A
C
∆𝐴𝐷𝐵~ ∆𝐵𝐷𝐶 ~ ∆𝐴𝐵𝐶
2.
Pythagorean Theorem
c
a
a 2  b 2  c2
Kinds of triangles (note c is always the longest side)
2
2
2
If a  b  c then ∆ is a right triangle
2
2
a  b  c 2 then ∆ is an acute triangle
a 2  b 2  c 2 then ∆ is an obtuse triangle
b
1
3.
Families of right triangles
(3, 4, 5)  (6, 8, 10)  (9, 12, 15)
4.
Most common seen triples
(5, 12, 13) (8, 15, 17) (7, 24, 25) (9, 40, 41)
5.
Special Right Triangles
45°
6. Trigonometry - SOH - CAH - TOA
opposite
a
sin  =

hypotenuse c
60°
x
2x
x
45°
30°
c
a
adjacent
b
cos  =

hypotenuse c
b
x
opposite a
tan  =

adjacent b
B
7.
Law of Sines
a
b
c


sin A sin B sin C
A
8. Law of Cosines
a
c
B
C
b
a 2  b 2  c 2  2bc cos A
b 2  a 2  c 2  2ac cos B
c 2  a 2  b 2  2ab cos C
a
c
A
b
C
COORDINATE GEOMETRY
Given AB where A ( x 1 , y1 ) B ( x 2 , y 2 )
x 2  x12  y2  y12
1.
distance:
2.
 x1  x 2 y1  y2 


,
midpoint:  2
2 
3.
slope:
a. m  y2  y1
x2  x1
b.
4.
parallel lines
c. perpendicular lines
Equation of a line
y  mx  b
opposite reciprocal slope
another way
slope-intercept
a.
same slope
or
point slope form
y  y1
b.
m
x  x1
or c.
y  y1  m  x  x1 
POLYGONS (n = number of sides)
1.
Sum of measures of interior angles
2.
Sum of the measure of exterior angles
3.
Number of diagonals from one vertex
4.
5.
180˚ (n - 2)
360°
n-3
nn  3
Total number of diagonals
2
Regular polygon [all sides are equal, all interior angles are equal, all exterior angles are equal]
a.
b.
measure of one exterior angle 360
measure of one interior angle
n
180  n  2 
n
2
CIRCLES
A
1.
Angles
A
A
a.
b.
c.
B
1
B
1
1
B
central angle m1  mAB
inscribed angles m1 
d.
e.
C
tangent - chord angle m1 
B
1
E
1
A
A
C
1
B
Chord-chord angle

Secant-secant angle

1
m1  mAB  mCD
2
g.
1
m AB
2
C
f.
D
D
A
1
m AB
2
m1 
A

1
mDE  mBC
2
D
B
Tangent-secant angle

m1 

1
mCD  mAB
2

C
D
1
B
Tangent- tangent angle
1
m1  mACB  mADB
2

2.

Facts About Circles
B
D
C
b
B
a
O
A
c
d
A
a.
An angle inscribed in a semicircle
is a right angle. mD  90
b.
angles of inscribed quadrilateral
a  d  180 c  b  180
A
c.
radius drawn to point of
tangency is perpendicular
AB  BC
B
C
O
d.
The segment from the center of the circle to a chord
is perpendicular to chord AC  OB
3
3. Segments of the Circle; Power Theorems (create similar triangles)
D
E
A
B
B
D
E
C
D
a.
B
A
Chord - chord
AB  BC  DB BE
b.
C
A
C
Secant - secant
AB  AD  AC  AE
c.
Tangent - secant
AB2  AC  AD
Parallel Lines and Quadrilaterals
1
3
5
7
If p || q then,
4
6
8
2
p
q
t
1. Corresponding angles are congruent
1  5, 3  7, 2  6, 4  8
2. Alternate interior angles are congruent
3  6, 4  5
3. Alternate exterior angles are congruent
1  8, 2  7
4. Consecutive interior angles are supplementary
m3  m5  180
m4  m6  180
4
Quadrilaterals
Quadrilateral- 4 sided figure.
Sum of interior angles=360º
ParallelogramOpposite sides are parallel
Opposite sides are congruent
Opposite angles are congruent
Consecutive angles are supplementary
Diagonals bisect each other
TrapezoidOne pair of opposite sides are parallel
Isosceles TrapezoidOne pair of opposite sides are
parallel
Legs are congruent
Pairs of base angles are
congruent
Diagonals are congruent
Rhombusa parallelogram with 4 congruent sides
all properties of a parallelogram
diagonals are perpendicular
diagonals bisect opposite angles
Rectanglea parallelogram with 4 right angles
all properties of a parallelogram
diagonals are congruent
Squarea parallelogram with 4 right angles and 4 congruent sides
all properties of a parallelogram, rectangle and rhombus
5
AREAS
1.
Rectangle
2. Parallelogram
A=b•h
h
b
3.
Triangle
4. Trapezoid
b1
h
h
b2
b
5.
Rhombus
6. Regular Polygon
D
B
A
C
a
a = apothem
p = perimeter
S
7.
Circle
r
Area = π r2
Circumference = π d
6
SURFACE AREA AND VOLUMES
1.
Right Prism
h
Lateral Area = (perimeter of base) (height)
Surface Area = Lateral area + 2 base areas
Volume = (Base area) (height)
w
l
2.
Cylinder
Lateral Area = 2 π r h
r
h
3.
 r2
V   r2  h
Surface Area = lateral area + 2
V = (Base area) (height) or
Pyramid
Lateral area =
1
(perimeter base) (slant height)
2
1
Pl
2
LA 
Surface area = lateral area + 1 base area
is the slant height
B is the area of the base
4.
V=
1
(Base area) (height)
3
Cone
Lateral area =
LA 
1
(perimeter base) (slant height)
2
1
 d 
2

Surface area = lateral area + 1 base area
TA 
1
 d    r 2
2
V = 1 (Base area) (height)
3
 
1
2
V   r h
3
5.
Sphere
Surface area = 4 r 2
4 3
r
Volume = 3
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