Leaving Cert Maths: Definitions Flashcards
Right angle
An angle that measures 90 degrees
Straight angle
An angle that measures 180 degrees
Obtuse angle
An angle that measures between 90 and 180 degrees
Reflex angle
An angle greater than 180 degrees (but less than 360)
Acute Angle
An angle that measures less than 90 degrees
Transversal Line
A line that intersects two or more lines
Alternate angles
Are on opposite sides of the transversal and between the intersected lines. Alternate angles between parallel lines are equal in size.
Corresponding angles
Angles which occupy the same relative position where a transversal crosses two other lines. If the two lines are parallel, the corresponding angles are equal.
Vertically opposite angles
Opposite angles formed when two lines cross. They are equal in size.
Isosceles triangle
A triangle which has two equal sides and two equal angles.
Equilateral triangle
A triangle which has three equal sides and three equal angles of 60 degrees.
Parallelogram
A four-sided polygon with both pairs of opposite sides being parallel.
Intersecting lines
Pass through the same intersection point
Perpendicular lines
Perpendicular lines intersect at an angle of 90 degrees
Complementary angles
Two angles that add up to 90 degrees
Congruent
Having the same size and shape
Supplementary angles
Angles whose measure adds up to 180 degrees.
Vertical angles
Angles opposite one another at the intersection of two lines. They are congruent ( the same size).
Vertex
A point where two or more straight lines meet.
Congruent angles
Angles that have the same measure
Acute angle
Angles whose measure is >0 degrees and <90 degrees
Degrees
A unit for measuring angles
Parallel lines
Lines (in the same plane) that never intersect.
Perpendicular Lines
Two lines that intersect to form right angles
Transversal
A line that intersects 2 or more lines
Corresponding Angles
Angles in the same place on different lines (they are equal in size if lines being cut are parallel)
Alternative Angles
Interior angles on the alternate sides of the transversal (they are both equal in size if lines being cut are parallel)
Co-interior Angles
Angles that are both inside the parallel lines and on the same side of the transversal. (They are supplementary)
incenter
point of concurrency of the 3 angle bisectors, it is equidistant from each side of the triangle, it is the center of the incircle
circumcenter
point of concurrency of the 3 perpendicular bisectors, it is equidistant from each vertex of the triangle and is the centre of the circumcircle
circumcircle
circle passes through all 3 vertices of the triangle
orthocenter
point of concurrency of the 3 altitudes (perpendicular lines form vertices to opposite side)
centroid
point of concurrency of the 3 medians (line through the vertices and midpoints of the opposite side)
incircle
circle which has 3 sides of the triangle as tangents
Constant
A number that's value will never change.Ex. 2x+4 constant is 4
Variable (algebra)
A letter representing an unknown value.Ex. 2x+4 variable is x
Coefficient
Number in front of a variable, telling how many variables there are. Ex. 2x+4 coefficient is 2
Substitute
To replace a variable with an algebraic expression with a known value (eg. x=24) or an expression (eg. x=k+1)
Exponent
The top, little number which tells how many times the base is being multiplied eg.
Base
The big, bottom number which is the number being multiplied. eg.
Surd
A number left in square root form
Discriminant
The discriminant is the part of the quadratic formula underneath the square root symbol: b^2-4ac. The discriminant tells us whether there are: -- two real solutions (two different roots)- one solution (two equal roots)- no solutions (two complex roots)
Absolute Value
The distance a number is from zero on a number line. ALWAYS POSITIVE, also called modulus
Expression
When there is a combination of constants, variables, and coefficients. Eg. 32x + 56
Equation
When two things are equal to each other (right side equal to left). eg. 24 = 12 + 12
Polynomial
It is an expression with variables that have positive whole number powersExampleDegree 1: y = 2x + 3 (linear)Degree 2: y = 3x^2 + 4x + 5 (quadratic)
Like Terms
Variables raised to the same power or exponent.Ex. 7c, -2c,+34c
Distributive property
Result obtained when a value is multiplied by another value eg. 2(3x - 5) = 6x - 10
Theorem
A rule that has been proved following a certain number of logical steps or by using a previous one or axiom that you already know
Example of a theorem
The angles in a triangle add up to 180 degrees
Axiom
A rule or statement accepted without any proof
Example of an axiom
There are 360 degrees in a full circle
Corollary
A statement that follows from a previous term
Example of a corollary
One theorem states that in a parallelogram, opposite sides are equal and opposite angles are equal. A ______of this is that a diagonal divides a parallelogram into 2 congruent triangles
Converse of a theorem
The reverse of a theorem
Example of a converse of a theorem
Theorem: if there are two equal angles in a triangle the the triangle is isoscelesConverse: if a triangle is isosceles hen there are 2 equal angles in the triangle
To imply
To use something we have proved previously
Example of imply
a + b = 15=> a = 15-b
Proof
A series of logical steps we use to prove a theorem
Proof by contradiction
A proof where we make an assumption, then prove it wrong by using valid axioms
Example of a proof by contradiction
Assume 2 angles in a triangle can both be 90 degrees90+90+c= 180c=0 This means that the third angle is 0 meaning that all 3 points would be collinear. Therefore by contradiction a triangle can't have 2 right angles
Is equivalent to
Something has the same measure as, or corresponds to, something else
Example of equivalent to
2/5 = 0.4
If and only if
when one thing is true only if another thing is true
Example of if and only if
A parallelogram is a rhombus if and only if all four of its sides have the same length
Similar triangle
A triangle where all the angles are equal - Equiangular triangle
Exterior angle
The sum of the 2 interior opposite angles
Congruent triangles
Triangles that are the same as each other1) SSS (side, side, side)2) SAS (side, angle, side)3) ASA (angle, side, angle)4) AAS (angle, angle, side)5) HL (hypotenuse in a right-angled triangle, leg)
Points about congruent triangles
1) their areas are equal2) if we can prove that 3 sides are equal, we can say that 3 angles are equal
Quadrilateral
Four sided figure
Properties of a square
1. Has 4 equal sides2. Opposite sides are parallel3. All angles are 90 degrees4. Diagonals bisect each other
Properties of a rectangle
1. Opposite sides are equal2. Opposite sides are parallel3. All angles are 90 degrees4. Diagonals bisect each other
Properties of a parallelogram
1. Opposite sides are equal2. Opposite sides are parallel3. Opposite angles are equal4. Diagonals bisect each other 5. The angles beside each other sum to 180 degrees
Properties of a rhombus
1. Has four equal sides2. Opposite sides are parallel3. Opposite angles are equal4. Diagonals bisect each other5. Angles beside each other add to 180 degrees
Similar triangles
Triangles that have equal corresponding angles and corresponding sides in the same ratio
Cyclic quadrilateral
A four sided shape that touches a circle at it's corner points
Three types of transformations
1. Translation2. Axial symmetry3. Central symmetry
Translation
A type of transformation that moves a figure in a straight line. It looks the same as before
Axial symmetry
A type of transformation that reflects a figure through a line
Central symmetry
A type of transformation that reflects a figure through a point
Enlargement
A type of transformation that changes the size of an object. The image created is similar to the object if their corresponding angles are equal
Scaling
A type of transformation that changes the size of an object. The image created is similar to the object if their corresponding angles are equal
Observational study
Researcher observes behaviour without influence
Designed experiment
A treatment is applied and a researcher observes the effects
Sample survey
Data is obtained from a sample of the population and is used to estimate attributes of the entire population
Simple random sampling
This gives each member of the population and equal chance of being chosen
Stratified random sampling
Population is divided into subgroups based on similar characteristics and a simple random sample is drawn from each subgroup depending on their proportion of the population
Systematic random sampling
A starting point is chosen at random and the sample is taken at regular intervals for example if you want to sample eight houses from 120, 15 is your interval, a random house between one and 15 is chosen at random and every 15th house from there is studied
Cluster sampling
Population divided into clusters and cluster is chosen at random
Quota sampling
Selection of the sample is made by interviewer who has quotas to fill in
Convenience sampling
Nonprobability sampling method. subjects chosen in most convenient way
Descriptive statistics
Use of graphs tables charts and various other measurements and calculations to organise and summarise information
Inferential statistics
A proportion of the population is taken and then conclusions are made about the entire population
Variable (statistics)
A numerical characteristic of interest in each element of the sample
Data set
Value of all observations of a variable for the elements of a sample
Data capture
Process by which data is transferred from a paper copy to an electric file
Observations
Value of the variable for one particular element of the sample
Univariate data
Only one piece of information is collected from each member of the group
Bivariate data
Two items of information are collected from each member of the group
Outlier
Very high or very low value not typical of the other values in the data set
Primary data
Data collected by the person who was going to use it
Secondary data
Data already available and not collected by the researcher directly themselves
Response variable
The variable whose changes we wish to study
Explanatory variable
The controlled variable whose effects on the response variable we wish to study
Control group
Group against which the effects of something are measured for example if testing for use of a drug this group would be given the inactive substance called the placebo
Quantitative data
Data that can be counted or measured
Qualitative data
Data that CANNOT be counted measured or answered in numbers
Questionnaires
A set of questions designed to obtain information from a group of people
Respondents
People who answer questionnaires
Discrete numerical data
Possible values are isolated points along a number line. (e.g. shoe size, age in years)
Continuous numerical data
Values could technically be any number between upper and lower limits. (e.g. weight)
Nominal categorical data
Categorical data that cannot be ordered (ranked) e.g. Hair colour
Ordinal categorical data
Categorical data that can be ordered (ranked) e.g. Grades: A, B, C...