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 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.

Congruent

Having the same size and shape

Alternative Angles

Interior angles on the alternate sides of the transversal (they are both equal in size if lines being cut are parallel)

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

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

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

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

Degrees

A unit for measuring angles

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.

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

Vertically opposite angles

Opposite angles formed when two lines cross. They 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

Quadrilateral

Four sided figure

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

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

Substitute

To replace a variable with an algebraic expression with a known value (eg. x=24) or an expression (eg. x=k+1)

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

Descriptive statistics

Use of graphs tables charts and various other measurements and calculations to organise and summarise information

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