## Circle Theorems 2

Tangents to circles

#### Investigation

### Theorem 1

The length of the tangents from a point to a circle are of equal length

### Theorem 2

The Angle between a tangent and a radius is a right-angle

### Theorem 3

The angel between a tangent and a cord is equal to the angle subtended by the cord

## Examples

Given that PA is a tangent find the value ofSince PA is a tangent

Given that PA and PB are tangents find the value ofSince PA and PB are tangents they must both be of equal length. Therefore is isosceles.base angles in an isosceles triangle are equal.the angle between a tangent and a radius is a right-angle.

Find the value ofSince

#### Questions

## Trigonometry: Right-angled triangles Quesitons

For each set of questions find the value of

Make sure to write down all the working. It is a good habit to be rigorous in your answers.

Set 1

Set 2

Set 3

Set 4

Set 5

## Trigonometry and Right-angled triangles

#### Conceptual Understanding

If we find the ratio between a pair of sides on a triangle the value will be a constant even on larger triangles provided the triangles are similar to each other.

### The right-angled trigonometry triangle

The hypotenuse is the longest side and is opposite the right-angle

The opposite is the side that is opposite the given angle. (Or the angle we wish to find)

The adjacent is the side next to the given angle. (It is between the given angle and the right angle)

### The ratios SohCahToa

When solving a trigonometry right-angled triangle problem we need two pieces of information and then use the relevant ratio to find the third.

There are three possible scenarios.

- We are finding an angle
- We are finding a side that is a numerator in the ratio
- We are finding a side that is a denominator in the ratio

## Examples

Type 1 (The unknown is the numerator)

We have the hypotenuse and we want the opposite side. The only ratio with hypotenuse and opposite in it is the sine ratio.

Let’s fill out the equation

Type 2 (The unknown is the denominator)

We have the opposite and we want the adjacent side. The only ratio with adjacent and opposite in it is the tangent ratio.

Let’s fill out the equation

Type 3 (The angle is unknown)

We have the hypotenuse and adjacent side. The only ratio with adjacent and hypotenuse in it is the cosine ratio.

Let’s fill out the equation

#### Questions

## Pythagoras’ Theorem Questions

Set 1: Find the value of a, b, c and d

Set 2: Find the value of a, b, c and d

Set 3: Find the value of a and c in the first diagram and the area of the second diagram

## Pythagoras’ Theorem Proof

### Consider a pair of right-angled similar triangles

Consider

Since it is right-angled that implies that

Which implies that is similar to the other two triangle.

Since the triangles are similar

Also

Substituting gives