# Archive | Geometry

## Proof That a Line From a Point to the Centre of a Circle Bisects the Angle Between Two Tangents Drawn From That Point

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,soand the triangles have three equal lengths so are congruent and angleandbisects

## Proof That Tangents From a Point to a Circle Are Equal in Length

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

## 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 […]

## Constructing the Locus of Points a Fixed Distance From a Given Line

To construct the set of points a fixed distance x from a line AB, with a set of compasses draw circles of radius x with centres at A and B, then draw tangents from one circle to another, parallel to the line. For example, construct the set of points 2 cm from the line AB […]

## Constructing a Triangle With Given Sides

Suppose we want to construct a triangle with sides 5 cm, 6 cm and 7 cm. We use compass and ruler. Start with the base, and mark a line of 5 cm, PQ below. Open the compass to a length of 6 cm and draw an arc as shown below. Now open the compass to […]

## Bearings

Bearings involve using trigonometry, generally the cosine or sine rules: Cosine Rule:  Sine Rule: For the above diagram, find a)The distance BC b)The bearing of A from B and the bearing of B from C. a)Label the triangle as above, with sides labelled by little letters opposite angles labelled by big letters. so b) We […]

## Transformations – Summary – Translations, Reflections, Rotations, Enlargements

The 4 basic transformations are illustrated above. Each translation movesin thedirection andin thedirection and is written  A rotation is defined by three things: clockwise or anticlockwise, the centre of rotation, written and the angle of rotation in degrees. A reflection is described by the line of reflection or the mirror line – we write down the […]