Maths - Geometry and Probability

Angles on a straight line add to 180.

Angles in a triangle add to 180.

Angles around a point add to 360.

The exterior angle of a traingle equals sum of opposite interior angles.

The sum of angles in any shape can be found using this formula where n is the number of sides:

(n - 2) x 180

• Alternate angles are equal.
• Co-interior angles are equal.
• Angles in a triangle or on a straight line add 180 degrees
• Angles in a quadrilateral or around a point add to 360 degrees
• Complementary angles add up to 90 degrees
• Supplementary angles add up to 180 degrees
• Corresponding angles add to 180 degrees
• Vertically opposite angles are equal

• Angle subtented at the centre is twice angle subtented at circumference
• Opposite angles of cyclic quadratic are supplementary
• Angles in the same segment are equal
• The angle subtended on the diameter of a circle is always a right angle
• Alternate segment theorem
• Any two tangents from a single point are the same length

Pythagoras' Theorem

a² + b² = c²

In this equation, "c" is the hypotenuse and "a" and "b" are the two other sides. You can also rearrange the equation to find other sides. Pythagoras' Theorem only works with right angled triangles and to find the length of the sides of the triangles.

c² - b² = a²

c² - a² = b²

SOH CAH TOA

Sinθ = opposite/hypotenuse

Cosθ = adjacent/hypotenuse

Tanθ = opposite/adjacent

Always rearrange and solve. To find a missing side, rearrange the equation so that there is the side you are looking for by itself on one side of the equation and then solve. To find a missing angle, divide the sides and multiply them by the inverse version of sin/cos/tan.

Two-way Tables:

A two-way table is a useful way to organise data that can be categorised by two variables. From this data, we can get the probability that these events will happen. Two-way tables always need total columsn and rows. 

Tree Diagrams:

Tree diagrams are useful to show the chance that events will happen together or the chance that two unrelated events will occur. If you are looking for an "AND" event, then you multiply down the tree. If you are looking for an "OR" event, then you will add down the tree. 

Sample Space Diagrams:

Sample space diagrams are useful to find what the probability of an event happening is, especially when rolling two dice. 

Variability in Results:

The true (or actual) probability of an event happening is unknown, for example the probability of getting a head on a coin. However, we know that the probability of getting a head on a coin is theoretically 50%. This is called the model probability and is determined based on the situation. In this case, this is one half. The margin of error (MoE) tells us the maximum distance that outcomes could be expected to be from the model probability in an experiment. The confidence interval (CI) tells us what the “reasonable” range of outcomes will be. The width of the confidence interval is equal to twice the margin of error and is symmetrical about the model probability. The margin of error is dependant on the number of trials (n) in the experiment. The larger the number of trials, the smaller the margin of error will be. The formula for the margin of error is: MoE = 1/√n. The expected number in an experiment with n trials and p model probability is: expected number = n x p.

Averages:

• The "mean" is where you add up all the numbers and then divide by the number of numbers.
• The "median" is the "middle" value in the list of numbers. To find the median, you need to list your numbers in order from smallest to largest. 
• The "mode" is the value that occurs most often. 
• The "range" of a list a numbers is just the difference between the largest and smallest values.

Box and Whisker Plots:

• Uses a numerical variable (quantitive) and a categorical variable (qualitative)
• Box plots include 5 vertical lines which stand for min, LQ, med, UQ and max
• --
• O - obvious (what do you see? what is the graph showing?)                                                     
• S - specific (give specifics, explain each obvious observation)                                                  
• E - evidence (back up each statement with evidence, like numbers or calculations)               
• M - meaning (meaning in the context, real world reasons)