## Circle theorem 1 - Angle at the centre

**Example:**

### 1

If an angle at the circumference of a circle measures 40° between two points, what is the corresponding angle at the centre formed by lines from these points to the centre?

## Circle theorem 2 - Alternate Segment

**Example:**

### 2

A circle has a tangent AB and a chord AC that meets at point A. If the angle between tangent AB and chord AC is 35 degrees, what is the angle in the alternate segment at point C?

## Circle theorem 3 - Perpendicular Chord Bisector

**Example:**

### 3

In a circle where a perpendicular from the centre (O) to a chord (PQ) divides it into two equal parts of 8 cm each, what is the full length of the chord (PQ)?

## Circle theorem 4 - Angle in a Semicircle

**Example:**

### 4

If X and Y are endpoints of a diameter and point Z is on the circumference to create a triangle (XYZ), what is this angle (XYZ)?

## Circle theorem 5 - Tangent-Secant

**Example:**

### 5

If a tangent from point A to a circle measures 5 units and the external part of a secant from A measures 2 units, with the entire secant measuring 7 units, what is the relationship according to the Tangent-Secant Theorem?

## Circle theorem 6 - Cyclic Quadrilateral

**Example:**

### 6

In a cyclic quadrilateral, if one of the opposite angles measures 80 degrees, what is the measure of the other opposite angle?

## Circle theorem 7 - Extended Chord

### 7

A person draws two secant lines from point P. The first secant has an external segment of 3 units and a total length of 9 units. The second has an external segment of 2 units and a total length of 8 units. Which calculation is correct?

## Circle theorem 8 - Chord-Chord Product

**Example:**

### 8

If chord AB intersects chord CD at point X within a circle and the segments are AX = 6 cm, XB = 4 cm, CX = 3 cm and XD = 8 cm, what is the product comparison?

## Circle theorem 9 - Equal Chord

**Example:**

### 9

If two chords, GH and IJ, are 8 cm long and the distance from the centre to chord GH is 2 cm, what is the distance from the centre to chord IJ?

## Circle theorem 10 - Radii to Tangents

**Example:**

### 10

If two tangents from point R touch a circle at points S and T and the radius to point S is 5 cm, what is the length of the radius to point T?

## Circle theorem 11 - Inscribed Angle

**Example:**

### 11

If an arc subtends 80 degrees at the circle's centre, what is the inscribed angle measurement that subtends the same arc?

## Final thoughts on geometry circle theorems

## Glossary

- Circle Theorems - Geometric rules describing the relationships between angles, lines and segments that intersect or form inside/around a circle.
- Tangent - A line that touches a circle, never entering the interior.
- Secant - A line that intersects a circle at two points, passing through the interior.
- Chord - A line segment with endpoints on the circle. It's entirely inside the circle.
- Cyclic Quadrilateral - A quadrilateral where all four vertices are on the circumference.
- Inscribed Angle - An angle formed by two chords in a circle that share an endpoint.
- Central Angle - An angle whose vertex is the circle's centre and whose sides pass through two points on the circle.
- Arc - A portion of the circumference of a circle.
- Radius - A line segment from the circle's centre to any point on its circumference.
- Diameter - A chord that passes through the centre of the circle, the longest possible chord.

*This post was updated on 05 Jul, 2024.*