To prove two triangles are congruent, it is sufficient two any of the following: The lengths of corresponding sides of each triangle are the same. If triangle 1 has lengths 4, 5 and 6 cm, and triangle 2 has lengths of 4, 5 and 6 cm, then the triangles are congruent. Each triangle has two […]

## Solving Equations Involving Algebraic Fractions

An algebraic fraction is any expression with fraction which include x terms in numerator and/or denominator. If this expression is equal to something then typically we can solve it by reararranging the fraction into a linear or quadratic equation and solving. For example, ifthen we can multiply both sides byto clear the fraction, obtaining Expanding […]

## The Regions on a Two Set Venn Diagram

There are four regions on a Venn diagram consisting of two sets, as shown below, wheremeans ‘in set A but not in set B’ etc and ℰ is the entire set of elements. We can however, combine these to obtain combinations as shown below, the shaded regions being defined by the text below each diagram. […]

## Proof of Formula for Chord Bisecting Diameter at Right Angles

For a chord bisecting a diameter of a circle at right angles as shown below, To prove it complete the triangle with one side as diameter to give the diagram below. The triangles andare similar, since both are right angled and angle If anglethen from triangle ABD,and from angle Equating these gives

## Completing the Square and Solving Quadratic Equations

Completing the square make it possible to find the maximum of minimum value of a quadratic function without sketching it., or to solve quadratic equations without using the quadratic formula.. We start with an expressionto express in the form We might multip[ly this out to obtainNow we equateto this and solve for the coefficients a, […]

## Transformations of Graphs

We can sort transformations of graphs into two types -x transformations or y transformations. Anything else is a combination of an x transformation followed by a y transformation or vice versa. A transformation is an x transformation if it is an argument of a function on the right hand side, or if it can be […]

## Composing Functions and Finding Inverses

Composing Functions Some functions can be extremely complicated, for example, It is often simpler to represent a single function as two separate functions, one carried out on the result of the first. For example, we are given the function(1) We could define the two functionsandand then To find we could use (1): Or we could […]

## Simultaneous Equations With One Equation a Quadratic

Simultaneous equations usually refers to equations of the form (1) (2) We solve these be equating the coefficients of or and then eliminating that term. For example, in this case we can multiply (1) by 3 to get then subtract (2) and from (1), Our problem here is to solve equations such as (1) (2) […]

## Quadrilaterals

A quadrilateral is a four sided shape. The internal angles for all quadrilaterals add up to 360 degrees. All shapes made up of four straight sides are quadrilaterals but some are special and are given special names. They are shown in the table below, along with their properties. Name Example Properties Square All the internal […]

## Calculating the Dimensions of Histogram Bars

A histogram is drawn as length of weight for example on the horizontal axis against frequency density on the vertical axis. Both axes have a linear scale. When drawing the bars of histograms there are three simple relationships: These equations may be used as in the below example: Notice that the intervals or bin sizes […]