- A chord is any straight line is a circle that joins any two parts of the circumference
- An isosceles triangle is formed by a chord and the two radii joining the ends of the chord to the centre of the circle

- This is not technically a circle theorem, but is very useful in answering circle theorem questions

- To start any circle theorem questions
- first identify any radii and mark them as equal lines
- then look to see if the radii are joined to any chords

- If a radius or diameter intersects a chord in a circle, in will
**bisect**that short at a right angle**bisect**means to cut in half

- This circle theorem is seen less often, but can be very useful in finding equal lengths and angles
- To spot it, look for a radius and see if it intersects any chords
- Problems involving this theorem often have the radii being joined to the end of the chords and so creating two congruent triangles
- This is also easier to see than remember from its description

- Although it is not strictly a circle theorem the following is a very important fact for solving some problems
- A
**triangle**which is formed from the centre using**a chord**and**two radii**is an**isosceles triangle**- This means at least
**two of the angles will be equal**and there will be at least**one line of symmetry** - This is very useful in
**proving circle theorems**

- This means at least

- A tangent to a circle is a straight line
**outside of the circle**that touches its circumference only**once** - Tangents are the easiest thing to spot quickly in a circle theorem question as they lie outside of the circle and stand out clearly

- Most of the time, if there is a tangent in a circle theorem question it will meet a radius at the point where it touches the circumference of a circle
- Make sure that the line the tangent meets is definitely a radius; that it starts at
*O*, the centre

- Make sure that the line the tangent meets is definitely a radius; that it starts at
- This circle theorem states that a radius and a tangent meet at 90°
- Perpendicular just means
**at right angles**

- Perpendicular just means
- If asked to state reasons and you use this theorem then use the key phrase;
- "A
**radius and a tangent meet at right angles**"

- "A

- Although it is not strictly a circle theorem the following is a very important fact for solving some problems
**Two tangents**from a circle to the**same point**outside of a circle are**equal length**- This means that a
**kite**can be formed by two tangents meeting a circle- Remember that a kite is essentially two
**congruent**triangles about its main diagonal - The kite will have two
**right angles**, where the tangents meets the radii

- Remember that a kite is essentially two

- If you spot a tangent on a circle diagram, look to see if it meets a radius and add in the right angle clearly to the diagram straight away
- In some cases just the act of doing this can earn you a mark!

Find the value of *θ *in the diagram below.

The lines *ST *and *RT *are both tangents to the circle and meet the two radii on the circumference at the points *S *and *T.*

Angle *TSO *= angle *TRO *= 90° (A radius and a tangent meet at right angles)

Use vertically opposite angles to find the value of the angle at *T* that is opposite the 25° angle.

Angle *RTS* = 25° (vertically opposite angles)

Mark these angles clearly on the diagram.

Angles in a quadrilateral add up to 360°. Use this to form an equation for θ.

Simplify.

Solve.

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