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Geometry, Probability, Stats, and Measurements Cheat Sheet by

A cheat sheet or test review for Grade 12 Geometry, Probability, Statistics, and Measurements. Colour-coded for organization Navy: Geometry Cyan: Measurements Red: Probability Magenta: Statistics

Geometry

Triangles
Are three sides, three angles, and all angles add up to 180 degrees.
Acute Triangles
All interior angles must be 0-90 degrees. All equila­teral triangles are acute.
Scalene Triangles
All sides and angles differ in measure.
Right Triangles
Only one angle is equal to 90 degrees
Isosceles Triangles
Two opposite sides and angles are equal to each other.
Equila­teral Traingles
All sides equal. All angles equal to 60 degrees.
Finding a missing internal angle:
a + b + c = 180°
50° + 30° + c = 180°
180° - 50° - 30° = c
100° = c

Straight lines are equal to 180 degrees.

Finding the exteri­or/­int­ernal angle with a straight line:
x + y = 180°
40° + y = 180°
180° - 40° = y
140° =y

Polygons

Polygons
Any enclosed geomet­rical shape that is composed of straight lines.
Regular Polygons
All sides and interior angles are equal.
Diagonals
A segment connecting two non-ad­jacent corners in a polygons.
Formula to find the sum of interior angles:
180°(n - 2). n = number of sides.
Formula to find the measure of interior angles:
(180°(n - 2))/n
Find the sum of interior angles of a nine (9) sided polygon.
180°(n - 2)
180°(9 - 2)
180°(7)
1260°

Find the measure of interior angles of a 3 sided polygon:
(180°(n - 2))/n
(180°(3 - 2))/3
(180°(­1))/3
180°/3
60°

Quadri­lat­erals

Quadri­lat­erals
Any four sided polygon.
Parall­elo­grams
Opposite sides are parallel to each other. Opposite sides and angles are equal in measure.
Rhombus
Parall­elo­grams with all sides that are equal.
Rectagles
Parall­elo­grams with opposite sides equal in measure. All angles equal to 90°.
Squares
Parall­elo­grams with all sides that are equal. All sides are 90°
Isosceles Trapezoids
One set of sides are parallel. Other sides equal in measure.
Kite
Two sets of equal sides. No lines are parallel.
Squares are also Rhombus, Rectan­gles, and Isosceles Trapezoids

Diagonals

Formula for finding the number of diagonals in a polygon:
D = (n(n-3))/2
Diagonals
- Cut parall­elo­grams into two equal triangles.
- Bisect each other.

Adjacent angles in a parall­elogram add up to 180°

Opposite angles are equal to each other.

Diagonal Diagram

Adjace­nt/­Opp­osite Angles Diagram

Same colours are opposite angles. Adjacent angles are next to each other.

Probab­ility

Probab­ility
The mathem­ati­cally likelihood that an event will occur. A ratio that compares the possible successful outcomes, to the total number of outcomes.
Probab­ility Formula
Number of successful outcomes, divided by total number of outcomes. (1/10)
Odds
A ratio that compares the number of possible successful outcomes to the number of possible unsucc­essful outcomes.
Odds Formula
Successful Outcomes : Unsucc­essful Outcomes
Theore­tical Probab­ility
A ratio that compares the number of possible successful outcomes to the total number of possible outcomes Determined by reason or calcul­ation.
Experi­mental Probab­ility
A ratio that compares the number of times an event occurs to the total number of trials or tests Determined by experi­ment.
Expected Value
Expected value is an applic­ation of probab­ility that involves the likelihood of a gain or loss.
Expected Value Formula
EV=[%(­gain) x $gain]­-[%­(lose) x $loss]
Probab­ility of picking card #5: 1/5
Odds of picking card #5: 1:4
Odds of not picking card #5: 4:1
Theore­tical Probab­ility: 1/5 chance of choosing card #5.
Experi­mental Probab­ility: He picked up card #5 two times. 2/5 of picking card #5.


There is a 1 in 5 chance of winning $4.00. It costs $1.00 to play.

EV=[%(­gain) $(gain)] - [%(loss) $(loss)]
EV=[1/5 4] - [4/5 1]
EV=[0.2 4] - [0.8 1]
EV=0.8 - 0.8
EV=$0
  

Law of Sines

Sine Law
Used to find lengths of sides, or angles of non-right triang­les. 
Formula:
a/sin(A) = b/sin(B) = c/sin(C)
Find side a:
a/sin(30°) = 15cm/s­in(45°)
a = sin(30­°)(­15c­m/s­in(­45°))
a = 10.61cm

Find sin(C):
sin(C)/9 = sin(47)/11
sin(C) = 9*[sin­(47­)/11]
C = sin-1(0.59838)
C = 36.75°

Find Side Diagram: Law of Sines

Find sin(C) Diagram

Law of Cosines

Cosine Law
Used to find angles or sides when Sine Law isn't possible.
Formula to find with a given angle:
a2 = b2 + c2 - 2bcCosA
Formula when there are no angles:
Cos(A) = (b2 + c2 - a2)/2bc
a/sin(40°) = 15/sin(B) = 8/sin(C) cannot be calculated so Cosine Law is used

Find side (a)
a2 = b2 + c2 - 2bcCosA
a2 = 152 + 82 - 2(15)(­8)C­os(40°)
a2 = 225 + 64 - 240 Cos(40°)
a2 = 105.14933
a = √105.14933
a = 10.25

Find cosine(A)
Cos(A) = (b2 + c2 - a2)/2bc
Cos(A) = (72 + 52 - 62)/2(7)(5)
Cos(A) = (49 + 25 - 36)/70
Cos(A) = 0.542857
A = cos-1 (0.542857)
A = 57.12°

Diagram: What to use

Measur­ement

Accuracy
Accura­cy of a measur­ement is how close the measur­ement is to the true value.
Precision
Precis­ion of measur­ements is how close they are to each other. The precision is determined by the number of decimal places.
Uncert­ainty
Uncert­ainty is the natural variation in measur­ements associated with instru­ments
Tolerance (∓)
The total amount that a measur­ement is allowed to vary. Add or subtract Tolerance to Nominal Value.
Nominal Value
The middle number that can be added or subtracted from to show the minimum or maximum value.
Tolerance: (Maximum Value - Minimum Value)/2
[Eg. (130-1­20)/2 = ∓5].
125 ∓ 5 = (125 - 5 = 120) or (125 + 5 = 130)
Tolerance can have different maximum and minimum values.
Eg. 125 (+5) (-3) = [125 + 5 = 130] or [125 - 3 = 122]

Measur­ement (conti­nued)

 
Nominal Value: Minimum Value + Tolerance
Eg. 120 + 5 = 125.

Precision: Lowest unit of measur­ement of the measuring device or the signif­icant decimal place.
87.32kg = 0.0>1<.

Uncert­ainty: Because not all measuring devices are accurate, you include an error with the measur­ement.
(Smallest Measure/2) Eg. 0.1/2 = ∓0.05

Central Tendency

Statistics
Is based upon data collected. From that, inferences and specul­ations are made. It is reliant upon the data and the interp­ret­ation of the data. 
Mean
The average of all data. The sum of all data, divided by the number of data.
Median
The set of values that is the middle of values arranged in ascending or descending order.
Even Median Formula
X[n/2] + X[(n/2)+1])/2. (n = number of values) (X = position of values)
Mode
The value that appears the most freque­ntly.
Outlier
A piece of data that is signif­icantly different from the rest.
5, 7, 8, 8, 8, 9, 10, 12, 13, 14, 15

Mean: (5+7+8­+8+­8+9­+10­+12­+13­+14­+15)/11 = 9.9 = 10
Median (Odd): Middle value = 9

5, 7, 8, 8, 8, 9, 10, 12, 13, 14, 15, 35

Median (Even): (X[12/2] + (X[(12/2­)+1]/2
= (X[6] + X[6+1])/2
= (10 + 12)/2
= 22/2
= 11
Mode: 8

Other Statis­tical Measur­ements

Range
The difference from the highest value to the lowest value in the data set.
Trimmed Mean
Removing the highest and lowest values and calcul­ating the mean so that data is accurately presented.
Weighted Mean
The average or mean of a data set in which each data point does not contribute an equal amount to the final average.
Weighted Mean Formula
Sum of the product of each item and its weight, divided by sum of the weightings
5, 7, 8, 8, 8, 9, 10, 12, 13, 14, 15, 35

Trimmed Mean: Remove 5 and 35. (7+8+8­+8+­9+1­0+1­2+1­3+1­4+1­5)/10 = 10.4, rounded up = 10

Weighted Mean: Will be in a diagram because I cannot figure out how to use cells.

Weighted Mean Diagram

Percen­tiles

Percen­tiles
A value below which a certain percent of the data falls.
Percentile Rank
A percentile rank of 50 (usually written P50) is the median because 50% (or half) of the values in the set are below the median value.
Percentile Rank Formula
P=(B/n) * 100. B: The number of scores below a given score, n: The number of scores. Always rounded to the nearest whole number
Stem Leaf Plot
A way to organize data in order of place value. The "tens digit and greate­r" is the stem and the "ones digit" is the leaf.
^
Will show on a diagram because I cannot figure out cells.
Ron scores 82% on his biology exam. A total of 200 students who wrote the same exam. 135 scored lower than Ron. What is Ron's percentile rank?

P=(B/n) * 100
P=(135/200) * 100
P=(0.675) * 100
P=67.5
P=68th Percentile Rank

Stem Leaf Plot Diagram

The "tens digit and greate­r" is the stem and the "ones digit" is the leaf.
                   
 

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  • Published: 9th February, 2024

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