# Archive | Geometry

## Pythagoras Theorem

With Pythagoras theorem we can find the lengths of a side in a right angled triangle given the other two sides. With the sides of the triangle as labelled above, we use the formula For example It is important to label the sides of the triangle a, b and c in the way shown above. […]

## The Cosine Rule

e can use the cosine rule to: 1.Find a side when it is opposite a known angle and that angle is between two known sides. 2. Find an angle when all three sides are known. 1. We start by labelling the sides a,b,c and the angles A,B,C with sides opposite their respective angles as shown. […]

## Circle Theorems – A Summary

The angle subtended at the centre of the circle is twice the angle subtended at the edge. The angle at the edge of a circle subtended by a diameter is a right angle. Opposite angles in a cyclic quadrilateral – where all the vertices are an the edge of the circle – add up to […]

## Trigonometry and Right Angled Triangles – Finding an Angle

There are three basic formulae, involving the ratio of the lengths of the sides. Given a right angled triangle, we first label the sides for the angle we choose or have been given: Given we have chosen the anglewe label the sides for this angle. Opposite = is the side opposite the angle  Hypotenuse =is the […]

## 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 […]

## Solving Cosine Equations for More Than One Solution

To solve the equationinvert both sides, obtaining This is not the only the solution. There are an infinite number of solutions. Typically though, we want solutions in the range 0° – 360°. We can use the cosine curve below to find the solutions. Draw the horizontal line y=0.4 (because the equation to solve is cos x […]

## Constructing an Angle of Thirty Degrees

We can construct an angle of 30° by first constructing an angle of 60° then bisecting it. To construct an angle of 60° start by drawing a line AB. Open the compass to the same length as AB and put the compass needle at B. Draw an arc as shown. Now put the compass needle at A […]

## When to Use the Cosine Rule and When to Use the Sine Rule

The Cosine Rule and the Sine Rule are both used to find sides and angles in triangles that are not right angled. For the triangle below, the Cosine Rule states The Sine Rule states Which rule to use depends on the combination of angles and sides you are given. Given three sides, use the Cosine […]

## Working With Column Vectors

Writing vectors as column vectors is more informative than using vector notation since more information is included, and is preferable when working in the x – y plane. Suppose we have the triangle below. P splits AC in the ratio 1:2. The vector from A to B isand the vector from B to C is  […]