## Circle Theorems in Geometry

**Radius**- A straight line from the centre of the circle to anywhere on its circumference.**Diameter**- A straight line through the centre of the circle, joining two points on its circumference. It's twice the length of the radius.**Chord**- A straight line segment where the endpoints lie on the circle.**Tangent**- A straight line that touches the circle at exactly one point.**Arc**- A part of the circumference of a circle.**Sector**- A part of a circle enclosed by two radii and an arc.**Segment**- A part of a circle is separated from the rest of the circle by a chord.

### 1

Which statement is true about the diameter of a circle?

## The Alternate Segment theorem

**The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.**

### 2

In a circle, if the angle formed by a tangent and a chord is 65 degrees, what is the angle in the alternate segment?

## Angle at the Centre Theorem

**The angle subtended at the centre of the circle by an arc is twice the angle subtended at the circumference by the same arc.**

### Breaking it down

### Practical implications

### 3

If an angle at the centre of a circle subtended by an arc is 120 degrees, what is the size of the angle subtended at the circumference by the same arc?

## Angles in the Same Segment theorem

**angles in the same segment of a circle are equal to each other.**

### 4

In a circle, if one angle in a segment formed by a chord is 45 degrees, what is the size of another angle in the same segment formed by a different chord?

## Angles in a Semicircle theorem

**any angle marked inside a semicircle is a right angle (90 degrees)**.

### Exploring the Angles theorem

### 5

If a triangle is inscribed in a semicircle with its base along the diameter, what is the measure of the angle at the apex?

## Chord of a Circle

**chords of equal length subtend equal angles at the centre of the circle**.

### Understanding Chord Properties

### 6

If two chords in a circle are of equal length, what can be said about the angles they subtend at the centre of the circle?

## Tangent of a Circle

**point of tangency**. A tangent is perpendicular to the radius at the point of tangency.

**if two tangents are drawn to a circle from an external point, they are equal in length**. This property highlights the symmetry in circle geometry and helps us to solve complex problems involving circles and lines.

### 7

If a line segment is drawn from an external point to touch a circle at one point, making a 90-degree angle with a radius at the point of contact, what is the line segment called?

## Cyclic Quadrilateral theorem

**the opposite angles sum up to 180 degrees (supplementary angles)**.

### Exploring Cyclic Quadrilaterals

### 8

Consider the angles in a cyclic quadrilateral. If one angle is 70 degrees, what is the measure of its opposite angle?

## Subtended Angles

**the angle subtended by an arc at the centre of the circle is twice the angle subtended by the same arc at any point on the circumference**.

### Understanding Subtended Angles

### 9

If an arc subtends an angle of 80 degrees at the centre of the circle, what is the measure of the angle subtended by the same arc at a point on the circumference?

## Conclusion

## Frequently asked questions

## Glossary

- Arc - A segment of a circle's circumference.
- Chord - A line segment with both endpoints on the circle's circumference.
- Cyclic Quadrilateral - A quadrilateral with all four vertices on the circumference of a circle.
- Diameter - A chord that passes through the centre of the circle, the longest possible chord.
- Radius - A line segment from the centre of the circle to any point on its circumference.
- Segment - A region of a circle bounded by a chord and the arc it subtends.
- Sector - A region of a circle bounded by two radii and the arc between them.
- Subtended Angle - An angle formed by two points on a circle's circumference and a third point, either on the circle or inside it.
- Tangent - A line that touches the circle at exactly one point.