The theorem is illustrated below. Proof: Construct the trianglesandby drawing radii as below. since both are radii of the circle andis common to both. Further, angle since these are between a tangent and a radius. From Pythagoras theorem,so

## Proof That the Angle Between a Chord and a Tangent at the Point of Contact is Equal to the Angle in the Alternate Segment

The theorem is illustrated below Proof: Draw a diameter atas below. The angleand angle Using the triangleanglethen by this theorem, Similarly angle

## Proof That Opposite Angles in a Cyclic Quadrilateral Add to 180 Degrees

The diagram below illustrates the theorem. and From C and B draw lines to the centre of the circle. Ifthenas below and if then so

## Proof That the Angle Subtended by an Arc at the Centre of a Circle is Twice the Angle Subtended at the Circumference

Triangleis isosceles sinceis isosceles similarly. We can labels the angles as below. Thenandas below. Then the remaining angles as below (sinceis a straight line, soand similarly for BOC) Then

## Frequency Polygons

Frequency polygons show the same information as bar charts, with points joined by lines instead of with vertical bars. Suppose we need to draw a frequency polygon for the data in the table below which contains information about the weights of some ambitious slimmers. Weight,in kg Frequency 12 35 76 80 41 17 The frequency […]

## Compounded Fractions and Percentages

Some fractions and percentage questions need to be read and well understood. Typical of these are questions that require you to find fractions or percentages of things and then fractions or percentages of the remainder. Example: If I start with £50, spend 20% of this on sweets and 30% of the remainder on a train […]

## Planes of Symmetry

A plane of symmetry cuts a shape in half so that on each side of the plane is a mirror image of the other side. Many shapes have planes of symmetry, including all prisms – shapes that have a constant cross section. This plane of symmetry of a prism is halfway along the prism, as […]

## Nets and Surface Areas

To find the surface area of a solid shape, we should often first draw the net of the solid. This involves ‘unfolding’ the solid onto a flat surface and finding the areas of individual parts of the net, which are often simple shapes like rectangles, squares, triangles and circles. To find the area of the […]

## Simplifying Algebraic Fractions

An algraic fraction is an fraction in which the numerator or denominator or both, include x terms. are all algebraic fractions. Apart from the last term, these fractions cannot be simplified. For an algebraic fraction to be simplified, the numerator or denominator or both must factorise so that there are common factors in numerator and […]

## Solving Equations Graphically

Suppose we sketch the graphWe obtain the graph below. We can use the graph to solve the equationby finding the intersection of the graph with the– axis, since this is whereSuppose we want to solve a different equation – for example,We could sketch the graph ofand find the intersection of the graph with the– axis. […]