Circle Geometry Definitions and Theorems for the ESAT

Basic Terminology of the Circle

Understanding the components of a circle is the first step in geometric problem solving. The following terms are essential for any student sitting the ESAT:

1. Circumference: The circumference of a circle, shown as the black outline in the diagram, represents the total distance around the outside of the shape.

2. Centre: The centre of a circle, marked as $O$, is the point which is the exact same distance from every point on the circumference. It is also the location where the lines of symmetry of the circle meet.

3. Radius: A line segment extending from the centre to any point on the circumference is known as the radius. This term also refers to the specific length of that line.

4. Diameter: This is a line segment that passes through the centre of the circle and has both endpoints on the circumference. The diameter is exactly twice the length of the radius.

5. Tangent: A tangent is a straight line that touches the circumference at a single point but does not cross or cut into the circle.

Arcs, Sectors, and Segments

Circles can be divided into smaller parts using curves and straight lines. These parts are classified as arcs, sectors, or segments.

An arc is any part of the circumference. A minor arc measures less than a semicircle, while a major arc measures more than a semicircle.

A sector is a region of a circle bounded by two radii and an arc. When two radii divide a circle into two regions of different sizes, the larger region is the major sector and the smaller region is the minor sector.

A chord is a straight line joining two points on the circumference. A chord divides the circle into two segments. If the segments are of different sizes, the larger is the major segment and the smaller is the minor segment.

The Relationship between Radius and Diameter

As shown in the diagram below, the diameter $AC$ consists of two radii, $OA$ and $OC$. Therefore, the length of the diameter is always twice the length of the radius.

Correct identification of these parts is vital for the exam. The following diagrams illustrate the standard labeling of a circle:

Circle Theorems: Angles at the Centre and Circumference

To solve ESAT problems, you must apply standard circle theorems concerning angles, radii, tangents, and chords. A key concept here is the term subtended. An angle can be subtended at a point by two other points, a line segment, an arc, or an object.

In the diagrams above, angle $ACB$ is the angle subtended at point $C$ by the arc $AB$ or the line segment $AB$.

The angle subtended at the centre of a circle by a chord is twice the angle subtended at the circumference by the same chord.

In these diagrams, the angle at the centre, $AOB$, is twice the angle at the circumference, $ACB$. Note that the direction is important: the theorem applies to the same chord $AB$. One diagram involves an obtuse angle $AOB$ while the other involves a reflex angle $AOB$.

Example: Angle at the Centre

Points $A$, $B$, and $C$ lie on a circle with centre $O$. The obtuse angle $AOB$ is $130^\circ$.

To find the marked angle $ACB$, we must identify the correct angle at the centre. The angle at the centre that is twice $ACB$ is the reflex angle $AOB$, not the obtuse $130^\circ$ angle.

1. Calculate the reflex angle: $360^\circ - 130^\circ = 230^\circ$.
2. Use the theorem: $230^\circ = 2 \times ACB$.
3. Angle $ACB = 230^\circ \div 2 = 115^\circ$.

The Angle in a Semicircle

When a diameter cuts a circle into two semicircles, the angle at the centre is $180^\circ$. Consequently, the angle subtended at the circumference in a semicircle is always $90^\circ$.

Example: Triangle in a Semicircle

$ABC$ is a triangle inscribed in a circle with centre $O$. $AC$ is a diameter and angle $BAC = 27^\circ$.

Since $AC$ is a diameter, angle $ABC$ must be $90^\circ$ because it is the angle in a semicircle. To find angle $BCA$, we use the fact that angles in a triangle total $180^\circ$:

Angle $BCA = 180^\circ - 90^\circ - 27^\circ = 63^\circ$.

Angles in the Same Segment

Angles subtended at the circumference by the same chord are equal, provided they are in the same segment.

In the diagram above, all angles subtended by chord $AB$ at the circumference in the same segment are identical. However, if the angles are in different segments, such as angles $x$ and $y$ in the diagram below, they are not equal.

Example: Using Multiple Theorems

In a circle with centre $O$, angle $ABE = 67^\circ$.

Identify which other angle must be $67^\circ$ and calculate angle $AOE$.

1. Angle $ABE$ is in the segment defined by chord $AE$. Angle $ADE$ is also subtended at the circumference by the same chord, so angle $ADE = 67^\circ$. Note that angle $ACE$ is not subtended at the circumference, so it does not equal $67^\circ$.
2. Angle $AOE$ is at the centre subtended by chord $AE$. Therefore, $AOE = 2 \times 67^\circ = 134^\circ$.

Alternate Segment Theorem

The angle between a tangent and a chord is equal to the angle subtended by that chord in the alternate segment.

Example: Alternate Segment Theorem

Triangle $BDE$ is inscribed in a circle with tangent $ABC$. Angle $BDE = 68^\circ$ and angle $EBC = 58^\circ$.

We can find angle $BED$ using two methods:

Method 1: Angle $ABD = 180^\circ - 68^\circ - 58^\circ = 54^\circ$ because $ABC$ is a straight line. Angle $ABD$ is between the tangent and chord $BD$. The alternate segment contains angle $BED$, so angle $BED = 54^\circ$.

Method 2: Angle $CBE$ is between the tangent and chord $BE$. The angle in the alternate segment is $BDE$, which is $58^\circ$. Using triangle $BDE$, angle $BED = 180^\circ - (68^\circ + 58^\circ) = 54^\circ$.

Angle between a Radius and a Tangent

The angle between a radius and a tangent at the point of contact is always $90^\circ$.

Example: Tangents from a Point

$ABC$ and $ADE$ are tangents at $B$ and $D$ to a circle centre $O$. Angle $BAD = 82^\circ$.

Angles $OBA$ and $ODA$ are both $90^\circ$ as they are angles between a radius and a tangent. In the quadrilateral $OBAD$, the sum of angles is $360^\circ$:

Angle $BOD = 360^\circ - (90^\circ + 90^\circ + 82^\circ) = 98^\circ$.

Properties of Cyclic Quadrilaterals

A cyclic quadrilateral is a four sided shape where all vertices lie on the circumference of a circle.

1. Both pairs of interior opposite angles add up to $180^\circ$.
2. The exterior angle is equal to the interior opposite angle.

Example: Cyclic Quadrilateral with Parallel Lines

$ABCD$ is a cyclic quadrilateral with $BC$ parallel to $AD$. $BAE$ is a straight line.

Method 1: The exterior angle $DAE$ is equal to the interior opposite angle $BCD$, which is $74^\circ$. Because $BC$ and $AD$ are parallel, angle $CBA$ is equal to the corresponding angle $DAE$, so $CBA = 74^\circ$.

Method 2: Since $BC$ is parallel to $AD$, angle $CDA = 180^\circ - 74^\circ = 106^\circ$ (co-interior angles). In the cyclic quadrilateral, opposite angles add to $180^\circ$, so $CBA = 180^\circ - 106^\circ = 74^\circ$.

Combining Circle Theorems

Most ESAT questions require multiple theorems. Consider a tangent $ABC$ touching a circle at $B$, where angle $DBC = 58^\circ$.

To find angle $BOD$, first locate the angle in the alternate segment. Draw lines $BE$ and $DE$ to a point $E$ on the circumference.

1. Angle $BED = 58^\circ$ (alternate segment theorem).
2. Angle $BOD = 2 \times 58^\circ = 116^\circ$ (angle at the centre is twice the angle at the circumference).