Geometry
Notes and Assignments
EOC Review
Formulas, Postulates, Theorems, Properties, ect…:
A. Segments (2-D):
M is the midpoint of AB , iff. AM ≅ MB
B. Segments (3-D): A ( x A , y A ,z A ) and B ( xB , yB ,z B )
1. Distance: AB =
(( x
− x A ) + ( yB − y A ) + ( z B − z A )
2
B
2
⎛ ( x + x ) ( y + yB ) ( z A + z B ) ⎞
2. Midpoint: M ⎜ A B , A
,
⎟
2
2
2
⎝
⎠
3. In a 3-D coordinate system:
• x-axis: front and back
• y-axis: right and left
• z-axis: up and down
• point: A ( x, y,z )
2
)
C. Angles:
7. Def. Angle Bisector: BD bisects ∠ABC , iff. ∠ABD ≅ ∠DBC .
D. Properties of Equality or Congruence:
1. Addition Property: If AB = CD , then
AB + BC = BC + CD .
3. Multiplication Property: If AB = CD ,
then ki AB = ki CD
5. Reflexive Property: AB = AB ,
∠A ≅ ∠B , AB = AB , m∠A = m∠B .
2.
Subtraction Property: If
AB + BC = BC + CD , then AB = CD .
4.
Division Property: If ki AB = ki CD ,
then AB = CD .
6. Symmetric Property: If AB ≅ CD , then
CD ≅ AB , etc…
7. Substitution Property: If AB = CD , and y = f ( AB ) , then y = f ( CD ) .
Example: If AB = CD , and PQ = AB + MN , then PQ = CD + MN .
Or if m∠A = m∠B + m∠C , and m∠D = 2i m∠A , then m∠D = 2i ( m∠B + m∠C )
8. Transitive Property: If AB = CD , and PQ = CD , then AB = PQ , or if ∠A ≅ ∠B , and
∠P ≅ ∠B , then ∠A ≅ ∠P , etc…
9. Proportion Properties:
AB PQ
AB PQ
a. Cross Product: If
, then b. Exchange: If
, then
=
=
CD MN
CD MN
ABi MN = CDi PQ .
MN PQ
AB CD
=
, and
.
=
PQ MN
CD AB
E. Parallel Lines and Transversal:
F. All Triangles:
7. The larger angle in a triangle is opposite the larger side.
8. The sum of the lengths of any two sides of a triangle is greater than the length of the
remaining side.
G. Isosceles and Equilateral Triangles:
Note: In each of the
diagrams to the left,
C is called “the”
vertex angle and:
m∠C = 1800 − 2m∠A
m∠C = 1800 − 2m∠B
Equilateral Triangle
= 600, 600, 600.
3. In an Equilateral Triangle, each of the three angles measures 600.
H. Right Triangles:
1. Pythagorean Theorem: leg12 + leg 2 2 = hyp 2 or
opp 2 + adj 2 = hyp 2
2. Trigonometry:
“Find a side:”
opposite
.
adjacent
opposite
b. Sine, sin (θ ) =
.
hypotenuse
adjacent
c. Cosine, cos (θ ) =
.
hypotenuse
“Find an angle:”
⎛ opposite ⎞
d. Arctangent, tan −1 ⎜
⎟ =θ .
⎝ adjacent ⎠
a. Tangent, tan (θ ) =
⎛ opposite ⎞
e. Arcsine, sin −1 ⎜
⎟ =θ .
⎝ hypotenuse ⎠
⎛ adjacent ⎞
f. Arccosine, cos −1 ⎜
⎟ =θ .
⎝ hypotenuse ⎠
4. 45-45-90:
3. 30-60-90:
5.
Altitude to Hypotenuse:
I. Proving Congruent Triangles: (Note: AAA and ASS do not prove triangles congruent)
J. Using Congruent Triangles: CPCTC (Corresponding Parts of Congruent Triangles are
Congruent). This means that if ABC ≅ PQR , then ∠A ≅ ∠B, ∠B ≅ ∠Q, ∠C ≅ ∠R , and
AB ≅ PQ, BC ≅ QR, CA ≅ RP.
K. Proving Similar Triangles:
Note: In congruent triangles, all of the
corresponding sides and angles are congruent
(CPCTC), but in similar triangles only the
corresponding angles are congruent. The
corresponding sides are proportional.
AB BC CA
ABC ~ PQR →
=
=
PQ QR RP
L. Using Similar Triangles:
M. Convex Polygons (Angles):
1. Angles (interior):
a. Sum = ( n − 2 ) 180 0
b. One ( regular only )
n − 2 ) 1800
(
=
n
2. Exterior Angles:
a. Sum = 3600
3600
b. One ( regular only ) =
n
3. Central Angles:
a. Sum = 3600
3600
b. One ( regular only ) =
n
N. Quadrilaterals:
1. Parallelogram: Opposite sides are parallel (bi-conditional).
a. Opposite sides are congruent (bi-conditional).
b. Opposite angles are congruent (bi-conditional).
c. Diagonals bisect each other (bi-conditional).
d. One diagonal divides it into two congruent triangles (bi-conditional).
2. Rectangle: Parallelogram with right angles (bi-conditional).
a. Congruent Diagonals (bi-conditional).
3. Rhombus: Parallelogram with congruent consecutive sides (bi-conditional).
a. Perpendicular diagonals (bi-conditional).
b. Diagonals bisect the vertices (bi-conditional).
4. Square: Definition: A rectangle and a rhombus
5. Trapezoid: One and only one pair of parallel “bases”.
(b + b )
a. midsegment = 1 2
2
b. Isosceles Trapezoid: Congruent legs
1) Congruent base angles
2) Congruent diagonals
O. Perimeter, and Area: ( P = perimeter , A = area , b = base ,
h = height or altitude or distance between bases , n = number of sides , a = apothem ,
l = leg , slant or lateral height , d = diagonal or diameter , m = midsegment , r = radius ,
C = circumference , L = arc length , θ = angle or arc measure , π 3.14 )
1. Parallelogram: P = 2b + 2l , A = bh
a. Rectangle: P = 2b + 2h , A = bh
dd
b. Rhombus: P = 4b , A = bh or A = 1 2
2
2
c. Square: P = 4b , A = b
(b + b )
2. Trapezoid: P = b1 + b2 + l1 + l2 , midsegment = 1 2 , A = mh
2
bh
3. Triangle: P = a + b + c , A =
2
b2 3
a. Equilateral Triangle: P = 3b , A =
2
( leg1i leg 2 )
b. Right Triangle: A =
2
Pa
.
4. Regular Polygon: P = nb , A =
2
5. Circle: C = 2π r , A = π r 2
a. Arc Length and Area of Sector (part of a circle): L =
b. Ring bound by
P and
Q , where
P. Polyhedra: (pictured on page 387 of text)
1. Tetrahedron: 4 equilateral triangles
2. Hexahedron: 6 squares
3. Octahedron: 8 equilateral triangles
4. Icosahedron: 20 equilateral triangles
5. Dodecahedron: 12 regular pentagons
θ
0
i 2π r , A =
360
P > Q : A = π rP 2 − π rQ 2 .
θ
360
0
iπ r 2
Q. 3-D Surface Area and Volume: ( PB = base perimeter , BA = base area ,
h = height or altitude or distance between bases , n = number of lateral sides ,
l = slant or lateral height , d = diagonal or diameter , r = radius , C = circumference ,
LA = arc length , S A = total surface area , V = volume )
1. Prism: LA = PB i h , S A = 2BA + PB i h , V = BAi h
2. Cylinder: LA = 2π ri h , S A = 2π r 2 + 2π ri h , V = π r 2 i h
P il
P il
B ih
3. Pyramid: LA = B , S A = BA + B , V = A
2
2
3
2
πr ih
4. Cone: LA = π ri l , S A = π r 2 + π ri l , V =
3
3
4π r
5. Sphere: S A = 4π r 2 , V =
3
R. Circles (Tangents, Secants, and Chords) (See length, perimeter, and area formulas for arcs)
1. Angle Formulas
2. Segment Formulas
3. Related Circles
S. Transformation Rules:
1. Translations right/left a and up/down b :
( x', y' ) = ( x ± a, y ± b ) ,
⎡ x1 x2 x3 ⎤ ⎡ ± a ± a ± a ⎤ ⎡ x1 ± a
⎢
⎥+⎢
⎥=⎢
⎣ y1 y2 y3 ⎦ ⎣ ±b ±b ±b ⎦ ⎣ y1 ± b
2. Reflections:
a. Across the x-axis:
( x', y' ) = ( x, − y ) ,
x2 ± a
y2 ± b
⎡ 1 0 ⎤ ⎡ x1
⎢0 −1⎥i ⎢ y
⎣
⎦ ⎣ 1
⎡ −1
⎢ 0
⎣
⎡0
c. Across the line y = x: ( x', y' ) = ( y,x ) , ⎢
⎣1
b. Across the y-axis:
( x', y' ) = ( − x, y ) ,
d. Across the line y = -x:
( x', y' ) = ( − y, − x ) ,
3. Rotations:
a. 900 clockwise around the origin:
x3 ± a ⎤
.
y3 ± b ⎦⎥
x2
y2
x3 ⎤ ⎡ x1
=
y3 ⎥⎦ ⎢⎣ − y1
x2
− y2
x3 ⎤
− y3 ⎥⎦
0 ⎤ ⎡ x1 x2 x3 ⎤ ⎡ − x1 − x2 − x3 ⎤
=
i
y2
y3 ⎥⎦
1⎥⎦ ⎢⎣ y1 y2 y3 ⎥⎦ ⎢⎣ y1
1⎤ ⎡ x1 x2 x3 ⎤ ⎡ y1 y2 y3 ⎤
=
i
0 ⎥⎦ ⎢⎣ y1 y2 y3 ⎥⎦ ⎢⎣ x1 x2 x3 ⎥⎦
⎡ 0 −1⎤ ⎡ x1
⎢ −1 0 ⎥i ⎢ y
⎣
⎦⎣ 1
x2
y2
x3 ⎤ ⎡ − y1
=
y3 ⎥⎦ ⎢⎣ − x1
− y2
− x2
( x', y' ) = ( y, − x ) ,
y2
y3 ⎤
⎡ 0 1⎤ ⎡ x1 x2 x3 ⎤ ⎡ y1
⎢ −1 0 ⎥i ⎢ y y y ⎥ = ⎢ − x − x − x ⎥
⎣
⎦ ⎣ 1
2
3⎦
2
3⎦
⎣ 1
0
b. 90 counterclockwise around the origin: ( x', y' ) = ( − y,x ) ,
⎡0 −1⎤ ⎡ x1 x2 x3 ⎤ ⎡ − y1 − y2 − y3 ⎤
⎢ 1 0 ⎥i ⎢ y y y ⎥ = ⎢ x
x2
x3 ⎥⎦
⎣
⎦ ⎣ 1
2
3⎦
⎣ 1
c. 1800 clockwise or counterclockwise around the origin:
⎡ −1 0 ⎤ ⎡ x1
⎢ 0 −1⎥i ⎢ y
⎣
⎦⎣ 1
4. Dilations:
x2
y2
x3 ⎤ ⎡ − x1
=
y3 ⎥⎦ ⎢⎣ − y1
− x2
− y2
( x', y' ) = ( − x, − y ) ,
− x3 ⎤
− y3 ⎥⎦
⎡ x1
⎣ y1
( x', y' ) = k ( x, y ) = ( ki x + ki y ) , ki ⎢
x2
y2
x3 ⎤ ⎡ ki x1
=
y3 ⎥⎦ ⎢⎣ ki y1
ki x2
k i y2
ki x3 ⎤
ki y3 ⎥⎦
− y3 ⎤
− x3 ⎥⎦
T. Other Formulas and truths worth noting:
1. Scale Factor (Ratio of Linear Measurement): k =
Im age
Pr e − image
Area of Im age
= k2
Area of Pr e − Im age
Volume of Im age
3. Ratio of Volumes:
= k3
Volume of Pr e − Im age
2. Ratio of Areas:
4. Square of a Binomial:
5. Quadratic Formula:
( x + y)
2
= x 2 + 2xy + y 2
( −B ±
Ax + Bx + c → x =
2
(B
2
− 4 AC )
( 2 A)
)
6. Probability: The ratio of the area (or volume) of the figure you are trying to hit to the
area (or volume) of the figure you could possibly hit.
Desired Area
Part ⎞
⎛
Pr obability =
, ⎜ AKA P =
⎟.
Total Possible Area ⎝
Whole ⎠
7. Conditional Statement: If A then B.
8. Converse: If B then A. (Change the order.)
9. Inverse: If opposite of A then opposite of B. (Change the signs.)
10. Contrapositive: If opposite of B then opposite of A. (Change the order and the signs.)
• Note: The original conditional is true, iff. the contrapositive is true. The original
conditional is false, iff. the contrapositive is false.
11. Logical Chain: If A then B. If B then C. If C then D. Conclusion, If A then D.
12. A square is divided diagonally into two 45-45-90 triangles.
13. Rotating a point around a given point creates a circle.
14. Rotating a rectangle around one side creates a cylinder.
15. Rotating a triangle around one side creates a cone.
16. Rotating a circle around its diameter creates a sphere.
17. All radii of a circle or all radii of a sphere are congruent.
U. Other Things to Remember:
1. Always draw a diagram. Many diagrams will involve right triangles.
2. Always re-read the question before recording your final answer. Sometimes the problem
asks you to “find x,” but often it asks you to find a measurement that requires you to plug
x back into an expression.
3. Follow the order of operations, including parentheses when necessary. Cancel in reverse
order of operations when solving for a variable (such as x).
4. Put a 1 under any whole expression when necessary to create a proportion, then cross
multiply.
5. When the variable is on top of a proportion, multiply. When the variable is on the bottom
of a proportion, divide. This occurs in many trigonometry problems.
6. Many problems can be solved by forming a right triangle and using either the
Pythagorean Theorem or trigonometry.
7. Whenever possible, use the length of the radius to form a right triangle and solve circle or
sphere problems using the Pythagorean Theorem.
8. Whenever possible, substitute each answer choice into the problem to determine which
answer is the “best choice.”
V. Your Notes (Use the space below to write any other notes that may help you on the exam):
Assignments:
1. EOC Sample Items
2. To be determined by needs revealed by evaluation of assignment 1.