Leaving Cert Maths: Definitions
Properties of a rhombus
1. Has four equal sides 2. Opposite sides are parallel 3. Opposite angles are equal 4. Diagonals bisect each other 5. Angles beside each other add to 180 degrees
Properties of a rectangle
1. Opposite sides are equal 2. Opposite sides are parallel 3. All angles are 90 degrees 4. Diagonals bisect each other
Properties of a parallelogram
1. Opposite sides are equal 2. Opposite sides are parallel 3. Opposite angles are equal 4. Diagonals bisect each other 5. The angles beside each other sum to 180 degrees
Three types of transformations
1. Translation 2. Axial symmetry 3. Central symmetry
Example of equivalent to
2/5 = 0.4
Variable (algebra)
A letter representing an unknown value. Ex. 2x+4 variable is x
Transversal
A line that intersects 2 or more lines
Transversal Line
A line that intersects two or more lines
Surd
A number left in square root form
Constant
A number that's value will never change. Ex. 2x+4 constant is 4
Variable (statistics)
A numerical characteristic of interest in each element of the sample
Example of a proof by contradiction
Assume 2 angles in a triangle can both be 90 degrees 90+90+c= 180 c=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
Example of a theorem
The angles in a triangle add up to 180 degrees
Converse of a theorem
The reverse of a theorem
Exterior angle
The sum of the 2 interior opposite angles
Exponent
The top, little number which tells how many times the base is being multiplied eg.
Descriptive statistics
Use of graphs tables charts and various other measurements and calculations to organise and summarise information
Quadrilateral
Four sided figure
Congruent
Having the same size and shape
Vertically opposite angles
Opposite angles formed when two lines cross. They are equal in size.
Example of imply
a + b = 15 => a = 15-b
circumcircle
circle passes through all 3 vertices of the triangle
incircle
circle which has 3 sides of the triangle as tangents
orthocenter
point of concurrency of the 3 altitudes (perpendicular lines form vertices to opposite side)
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
centroid
point of concurrency of the 3 medians (line through the vertices and midpoints of the opposite side)
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
If and only if
when one thing is true only if another thing is true
Alternative Angles
Interior angles on the alternate sides of the transversal (they are both equal in size if lines being cut are parallel)
Properties of a square
1. Has 4 equal sides 2. Opposite sides are parallel 3. All angles are 90 degrees 4. Diagonals bisect each other
Degrees
A unit for measuring angles
Points about congruent triangles
1) their areas are equal 2) if we can prove that 3 sides are equal, we can say that 3 angles are equal
Cyclic quadrilateral
A four sided shape that touches a circle at it's corner points
Parallelogram
A four-sided polygon with both pairs of opposite sides being parallel.
Example of if and only if
A parallelogram is a rhombus if and only if all four of its sides have the same length
Vertex
A point where two or more straight lines meet.
Proof by contradiction
A proof where we make an assumption, then prove it wrong by using valid axioms
Inferential statistics
A proportion of the population is taken and then conclusions are made about the entire population
Axiom
A rule or statement accepted without any proof
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
Proof
A series of logical steps we use to prove a theorem
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
Corollary
A statement that follows from a previous term
Designed experiment
A treatment is applied and a researcher observes the effects
Similar triangle
A triangle where all the angles are equal - Equiangular triangle
Equilateral triangle
A triangle which has three equal sides and three equal angles of 60 degrees.
Isosceles triangle
A triangle which has two equal sides and two equal angles.
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
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
Reflex angle
An angle greater than 180 degrees (but less than 360)
Straight angle
An angle that measures 180 degrees
Right angle
An angle that measures 90 degrees
Obtuse angle
An angle that measures between 90 and 180 degrees
Acute Angle
An angle that measures less than 90 degrees
Corresponding Angles
Angles in the same place on different lines (they are equal in size if lines being cut are parallel)
Vertical angles
Angles opposite one another at the intersection of two lines. They are congruent ( the same size).
Co-interior Angles
Angles that are both inside the parallel lines and on the same side of the transversal. (They are supplementary)
Congruent angles
Angles that have the same measure
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.
Supplementary angles
Angles whose measure adds up to 180 degrees.
Acute angle
Angles whose measure is >0 degrees and <90 degrees
Alternate angles
Are on opposite sides of the transversal and between the intersected lines. Alternate angles between parallel lines are equal in size.
Sample survey
Data is obtained from a sample of the population and is used to estimate attributes of the entire population
Polynomial
It is an expression with variables that have positive whole number powers Example Degree 1: y = 2x + 3 (linear) Degree 2: y = 3x^2 + 4x + 5 (quadratic)
Parallel lines
Lines (in the same plane) that never intersect.
Convenience sampling
Nonprobability sampling method. subjects chosen in most convenient way
Coefficient
Number in front of a variable, telling how many variables there are. Ex. 2x+4 coefficient is 2
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
Univariate data
Only one piece of information is collected from each member of the group
Intersecting lines
Pass through the same intersection point
Perpendicular lines
Perpendicular lines intersect at an angle of 90 degrees
Cluster sampling
Population divided into clusters and cluster is chosen at random
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
Data capture
Process by which data is transferred from a paper copy to an electric file
Observational study
Researcher observes behaviour without influence
Distributive property
Result obtained when a value is multiplied by another value eg. 2(3x - 5) = 6x - 10
Quota sampling
Selection of the sample is made by interviewer who has quotas to fill in
Is equivalent to
Something has the same measure as, or corresponds to, something else
Base
The big, bottom number which is the number being multiplied. eg.
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
Substitute
To replace a variable with an algebraic expression with a known value (eg. x=24) or an expression (eg. x=k+1)
Example of a converse of a theorem
Theorem: if there are two equal angles in a triangle the the triangle is isosceles Converse: if a triangle is isosceles hen there are 2 equal angles in the triangle
Example of an axiom
There are 360 degrees in a full circle
Simple random sampling
This gives each member of the population and equal chance of being chosen
To imply
To use something we have proved previously
Congruent triangles
Triangles that are the same as each other 1) 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)
Similar triangles
Triangles that have equal corresponding angles and corresponding sides in the same ratio
Complementary angles
Two angles that add up to 90 degrees
Bivariate data
Two items of information are collected from each member of the group
Perpendicular Lines
Two lines that intersect to form right angles
Data set
Value of all observations of a variable for the elements of a sample
Observations
Value of the variable for one particular element of the sample
Like Terms
Variables raised to the same power or exponent. Ex. 7c, -2c,+34c
Outlier
Very high or very low value not typical of the other values in the data set
Equation
When two things are equal to each other (right side equal to left). eg. 24 = 12 + 12
Expression
When there is a combination of constants, variables, and coefficients. Eg. 32x + 56
