FORMULAS
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The radius of a circle inscribed in a triangle is the area of the triangle divided by half of
its perimeter (semiperimeter).
- Or 2 areas/a+b+c
● To find the area of the triangle… use the Heron’s formula for the area of a
triangle that’s not a right triangle● Area = 𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑏)
Inscribed radius =
𝑠(𝑠−𝑎)(𝑠−𝑏)(𝑠−𝑐)
𝑠
s= the semiperimeter
a=side length 1
b=side length 2
c=side length 3
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The formula of a circle is written as
(x-a)²+(y-b)²=r²
r=radius of the circle
(a, b) is the coordinates of the center of the circle
Point: plug in a point on circle and it works
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Variations on the form distance divided by time=speed
D/T=V
D/V=T
V•T=D
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IMPORTANT THINGS TO REMEMBER
1
(n+1)² = n²+2n+1
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ARITHMETIC SEQUENCE
is a sequence where the difference between consecutive numbers are constant
Ex:
x, x+1,x+2, x+3, x+4
HOW TO FIND A NUMBER IN AN ARITHMETIC SEQUENCE
4, 7, 10, 13
find the 500th #
3N+1 (for this particular sequence)
N=# in sequence
If n=500, the 500th number is
3(500)+1
1500+1
1501
HOW TO FIND THE SUM OF AN ARITHMETIC SEQUENCE
Find the sum of
9, 16, 23… 72
2
7n + 2
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N-sequences and Triangular Numbers
Triangular numbers- 1+2+3+4+...n=
𝑛(𝑛+1)
2
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Inequalities
3
Shape Similarities
There are four theorems that we can use to determine if two triangles are similar.
AA theorem
Two triangles are similar if their two corresponding angles are congruent.
SSS theorem
Two triangles are similar if the lengths of all corresponding sides are proportional.
SAS theorem
4
Two triangles are similar if the corresponding lengths of two sides are proportional and
the included angles are congruent.
SSA theorem
Two triangles are similar if the lengths of two corresponding sides are proportional and
their corresponding angles across the larger of these two are congruent.
PLEASE SEE https://www.mathemania.com/lesson/triangle-similarity/ for more
information.
-If Triangles ABC and abc are similar
Any side of ABC/any side of abc=x
Area of ABC/area of abc
=x ²
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Sin, Cos, Tan
Sine, Cosine and Tangent (often shortened to sin, cos and tan) are each a
ratio of sides of a right angled triangle:
5
Here
a: Opposite of Ө
b: Adjacent of Ө
c: Hypotenuse
sinӨ = a/c = opposite/hypotenuse
cosӨ = b/c = adjacent/hypotenuse
tanӨ = a/b = opposite/adjacent
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◦
1
2
◦
2
2
; 𝑐𝑜𝑠45 =
◦
3
2
; 𝑐𝑜𝑠60 =
(𝑠𝑖𝑛30 =
(𝑠𝑖𝑛45 =
(𝑠𝑖𝑛60 =
◦
; 𝑐𝑜𝑠30 =
3
2
◦
◦
2
2
1
2
◦
; 𝑡𝑎𝑛30 =
3
3
)
◦
; 𝑡𝑎𝑛45 = 1)
◦
; 𝑡𝑎𝑛60 = 3)
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Important triangles: 30-60-90s and right
triangles
Relationship of 30-60-90
Find the sides opposite the 30-60-90 angles
The relationship is1 − 3 − 2
-------------------------------------------------------𝑘(𝑘+1) 2 𝑘(𝑘−1) 2
𝑘=
2
2
For example
3
3
(
2
)(
)
2
1 =1 -0
7
3
2
2
3
2
2
2 =3 -1
3 =6 -3 ….
Ptolemy’s Theorem
For a geometric sequence with first term a and common ratio r, the sum of
the first n terms is given by:
𝑛
𝑛
∑ 𝑎𝑖 = 𝑎
𝑖=1
1−𝑟
1−𝑟
____________________________________________________________________
𝑎
𝑏
·𝑎
𝑏
(𝑎 )
𝑐
𝑐
=𝑎
=𝑎
𝑏+𝑐
𝑏𝑐
where a is nonzero and b is positive =𝑎
−𝑏
=
1
𝑎
𝑏
-a to the -b= - 1/a to the b
__________________________________________
for a regular polygon…
a regular polygon is a polygon where it’s sides are equal in length
N= number of sides
Sum of the interior angles of a polygon = 180(n-2)
__________________________________________
1
𝑛(𝑛+1)
=
1
𝑛
1
- 𝑛+1
8
Ex: ½=1/1-½
⅙=½ - ⅓
Sum of the first n values of this form
1
𝑛(𝑛+1)
=
𝑛
𝑛+1
____________________________________________________________
SPHERES
Formulas you should remember:
4
3
volume =
π𝑟
3
surface area = 4π𝑟
2
______________________________________________
SQUARE
n
2
=𝑎
2
+𝑏
2
2
TRICK
−𝑐
2
a=
n-4
b= n/2 +3
c= n/2 - 5
Angles
9
Sum of all interior angles of a regular n-gon
Is (n-2)180
Sum of all exterior angles is
𝑛(𝑒𝑎𝑐ℎ 𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒 𝑑𝑒𝑔𝑟𝑒𝑒 𝑚𝑒𝑎𝑠𝑢𝑟𝑒) = 𝑛(180 −
180(𝑛−2)
𝑛
)=
180𝑛 − 180(𝑛 − 2)=360
Because the sum of all exterior 360, each exterior angle can also be written
as
360
𝑛
—————— — — - ——————- - —- — —--------------------------------------
Permutation vs Combination: what’s
the difference?
R*T=D: Clarification
10
11
Standard Form
How to get into Standard Form from Point-slope form
12
13
——— - — —- —————— ———
Slopes and Lines
● each line can be defined as it’s slope (what direction it’s headed) + a
point(s) it passes through
● So for 1.920, since you already know what point it passes through,
you find the slope in terms of the point (m= y/x). Then find # of
possible slopes within restrictions.
14
—----------------------------------------------------Shortcuts with Modular Arithmetic
15
Example: N100 ( mod a) -You can replace N with a number of equivalent
modulo, but you cannot replace the exponent with equivalent modulo
Binomial Theorem
Distance Formula/ Circle Formula (same
thing)
16
Circle formula
Points on circles (a,b)
Center of circle (x,y)
Radius=r
R2 = (a-x)2+(b-y)2
Important to understand they’re the exact same thing!!
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Vieta’s Formula
For 2nd Degree Polynomials - Proof for Quadratic Case
THE FORMULA
2
𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0 and roots 𝑟 and 𝑞, then the following is true:
𝑟+𝑠=
𝑟𝑠 =
𝑏
𝑎
−
𝑐
𝑎
PROOF
2
2
𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0, therefore 𝑥 +
𝑏
𝑎
𝑥+
𝑐
𝑎
=0
Since the roots are 𝑝, 𝑞 then the following is true
(𝑥 − 𝑝)(𝑥 − 𝑞) = 0
After expanding, we get:
2
𝑥 − (𝑝 + 𝑞)𝑥 + 𝑝𝑞 = 0
2
𝑥 +
𝑏
𝑎
𝑥+
𝑐
𝑎
=0
We also found out about this above.
So − (𝑝 + 𝑞) =
And
𝑏
𝑎
→ (𝑝 + 𝑞) =
𝑝𝑞 =
−
𝑏
𝑎
𝑐
𝑎
PROOF #2
By the Quadratic Formula - which is based on Completing the Square
18
For 3rd Degree Polynomials…and
Beyond!
3
2
𝑎𝑥 + 𝑏𝑥 + 𝑐𝑥 + 𝑑 = 0, and roots 𝑝, 𝑞, 𝑟, then the following is true:
𝑝+𝑞+𝑟=
−
𝑝𝑞 + 𝑞𝑟 + 𝑝𝑟 =
𝑝𝑞𝑟 =
−
𝑏
𝑎
𝑐
𝑎
𝑑
𝑎
PROOF
3
2
3
𝑎𝑥 + 𝑏𝑥 + 𝑐𝑥 + 𝑑 = 0 → 𝑥 +
𝑏
𝑎
2
𝑥 +
𝑐
𝑎
𝑥+𝑑=0
As the roots are 𝑝, 𝑞, 𝑟, we can rewrite the equation like this
3
𝑥 +
𝑏
𝑎
2
𝑥 +
𝑐
𝑎
𝑥 + 𝑑 = (𝑥 − 𝑝)(𝑥 − 𝑞)(𝑥 − 𝑟)
3
2
= 𝑥 − (𝑝 + 𝑞 + 𝑟)𝑥 + (𝑝𝑞 + 𝑝𝑟 + 𝑞𝑟) − 𝑝𝑞𝑟
Therefore:
𝑝+𝑞+𝑟=
−
𝑏
𝑎
19
𝑝𝑞 + 𝑞𝑟 + 𝑝𝑟 =
𝑝𝑞𝑟 =
−
𝑐
𝑎
𝑑
𝑎
A Generalization
How does Vieta’s work with (note: we first divided by 𝑎)?
𝑘
𝑘−1
𝑘−2
1
𝑥 + 𝑏𝑥
+ 𝑐𝑥
+... + 𝑚𝑥 + 𝑛
Answer: here’s the generalization
Let’s think about how we came up with the equation in the first place. We
had k solutions (roots) of 𝑥. Let 𝑟𝑖be the 𝑖
𝑥
𝑘
+ 𝑏𝑥
𝑘−1
+ 𝑐𝑥
𝑘−2
+... + 𝑚𝑥
1
𝑡ℎ
root of the equation.
+ 𝑛 = (+ 𝑥 − 𝑟1)(+ 𝑥 − 𝑟2)(+ 𝑥 − 𝑟3) ×... × (+
The writing of + 𝑥 was intentional. One thing you must not forget is that we
chose one item from each group [inside the ( _)]. This is quite similar to the
methodology behind the Binomial Theorem. The difference, however, is
that the binomial theorem relates the coefficients with the EXPANSION.
𝑛
(_) . Super-linear, the same thing multiplied over and over. With Vieta’s, it’s
different things being multiplied together. Vieta’s deals with the relationship
between coefficients and ROOTS. Vieta does not have hinge on something
being expanded, although it’s proof is based on a different type of
expansion. So really Vieta’s and the Binomial theorem are different types of
expansions.
With the differences explained, still remember we chose one item from
each group [inside the ( _)]. There are two choices, the root or the 𝑥.
And with this, we now can make our generalization (can be simplified, of
course).
𝑏 =− 𝑟1 +− 𝑟2 +− 𝑟3 + ... + − 𝑟𝑘
𝑐 = (− 𝑟1)(− 𝑟2) + (− 𝑟1)(− 𝑟3) +... + (− 𝑟𝑘−1)(− 𝑟𝑘)
…
20
𝑛 = (− 𝑟1)(− 𝑟2)(− 𝑟3)(− 𝑟4) ×... × (− 𝑟𝑘)
Note: whether 𝑘is odd or not only affects 𝑛, not the other coefficients
Final note: We did this with 𝑎 = 1, but for a real generalization, remember
that you need to divide by 𝑎 first! And then you have your generalization.
ADDITIONAL THINGS TO REMEMBER
An equation that comes up a lot with cubic Vieta’s.
21
——————————————----------------------------------------------------------
Exterior Angles for Quadrilaterals
Distinguishability
distinguishable= different (able to tell apart from each other)
indistinguishable= not different
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https://artofproblemsolving.com/wiki/index.php/Distin
guishability
1. Some motivating problems- AoPS
file:///C:/Users/tiech/Downloads/SMMATH_handout_distributions%20(2).pdf
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Slopes and Lines
24
25
————————————————————-
Intercepts
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Finding the Middle Number
26
Sequence: 1, 2, 3, … 65
Middle term:
1+65
2
= 33
Sequence: 1, 2, 3, … 98
Middle term: (98+1)/2= 49.5
**Special property of the Middle Term:
Sequence : 1 .... n
Middle term=x
x-1=n-x
Percents, Ratios, and the Comparison Trap
Example: Approximately 70.8% of Earth’s surface is covered with water;
the rest is land. It is said that Mars, whose surface is covered entirely with
land, has approximately the same amount of land as the earth. Based on
this information, what percent of Earth’s radius is Mars’ radius? Express
your answer to the nearest whole number. (2016 MathCounts State Team
#3)
what percent of Earth’s radius is Mars’ radius?
-pointing to EARTH’s radius- the base is EARTH’s radius.
- We’re comparing Mars's radius to EARTH’s radius
𝑀
𝐸
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27
Main Strategies in Number Theory
❖ Prime Factorization
❖ Modulo a number
❖ Factoring
---------------------------------------------------------------------------------------------------You can add (combine) rates when time is the same.
*You CANNOT combine rates when the time is not the same.
----------------------------------------------------------------------------------------------------
Optimization
Product of 2 numbers is the GREATEST when the numbers are closer
together.
Sum of 2 numbers is the GREATEST when … (actually the order doesn’t
matter for the sum)
Ex) AB + CD + EF = AD + CF + DB = 10(A+C+E) + (BDF)
Same is true for 2 digit numbers, 3 digits, etc. But the numbers have to be
in the same place (units, tens, etc.)!
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Fundamental Strategies in Finding Area or
Volume
❖ Split (Cutting): cut the shape into more familiar figures you
understand and know how to solve
❖ Embed: embed the funky shape into a more familiar figure
----------------------------------------------------------------------------------------------------
Good Angles and Bad Angles
❖ Good Angles:
➢ 30, 45, 60, 120, 150, 135
28
❖ Bad Angles:
➢ 15, 75, 105, 165
❖ Strategy:
➢ Split the Bad Angle to form good angles or
➢ Extend the Bad Angle (to get rid of it) to form perpendiculars or
Good Angles
Creating Similar Triangles
❖ Draw parallel lines
❖ Draw perpendicular lines
❖ Extend lines
❖ Make equal angles
29