Geometry Revision Notes

Key Definitions of Circle Parts

It's vital to know the precise definitions for different parts of a circle. The centre is the middle point. The radius (plural: radii) connects the centre to the edge. The diameter passes through the centre, connecting two points on the edge, and is twice the radius (d = 2r). The circumference is the total length around the circle's boundary. An arc is a part of the circumference. A chord is a straight line connecting two points on the circumference, while a tangent touches the circle at only one point.

Understanding Segments and Sectors

A sector of a circle is like a slice of pie. It's enclosed by two radii and the arc between them. Think of the hands of an analogue clock forming a sector. In contrast, a segment is a region of the circle enclosed by a chord and an arc. It's the area cut off by a chord. A segment does not include the centre of the circle, unless the chord is a diameter, in which case it forms a semicircle.

Recognising Lines Associated with Circles

Beyond the basic parts of the circle itself, there are specific lines associated with them. A tangent is a line that just 'kisses' the circle at a single point, forming a 90-degree angle with the radius at that point. A secant is a line that intersects the circle at two distinct points. While the term 'secant' may not be explicitly tested at this level, understanding its distinction from a tangent and a chord is helpful for a full grasp of circle geometry.

The Core Theorem

The fundamental relationship to remember is that the angle subtended by an arc at the centre of a circle is always twice the angle subtended by the same arc at any point on the remaining part of the circumference. This means if you have an angle at the centre, divide it by 2 to get the angle at the circumference. If you have an angle at the circumference, multiply it by 2 to get the angle at the centre. This theorem is valid for any arc, whether it's a major arc or a minor arc. Always identify the common arc that forms both angles.

Identifying the Correct Angles

It's crucial to correctly identify the angles involved. Both angles must be subtended by the *same* arc. For example, if points P and Q are on the circumference, and O is the centre, then angle POQ is the angle at the centre. If R is another point on the circumference not on the arc PQ, then angle PRQ is the angle at the circumference subtended by arc PQ. Make sure the 'arms' of both angles originate from the same two points on the circumference. Be careful with reflex angles; sometimes the angle subtended at the centre might be a reflex angle.

Applications and Extensions

This theorem has several important consequences. One common application is that the angle subtended by a diameter at any point on the circumference is a right angle (90 degrees). This is because the diameter subtends a straight angle (180 degrees) at the centre. Consequently, the angle at the circumference is half of 180 degrees, which is 90 degrees. This theorem also helps in understanding angles in cyclic quadrilaterals, where opposite angles sum to 180 degrees, as it can be derived from the centre-circumference relationship applied to different arcs.

The Angle in a Semicircle Theorem Defined

The Angle in a Semicircle Theorem states that an angle inscribed in a semicircle is always a right angle (90°). This applies when the vertices of the angle are on the circumference of the circle, and the side opposite to this angle is the diameter of the circle. This theorem is a direct consequence of the Angle at the Centre Theorem, which states that the angle subtended by an arc at the centre is twice the angle subtended by the same arc at any point on the remaining part of the circumference. For a semicircle, the arc subtends an angle of 180° at the centre (since it's a straight line, the diameter). Therefore, the angle subtended at the circumference by this same arc must be half of 180°, which is 90°.

Identifying a Semicircle Angle

To apply the Angle in a Semicircle Theorem, you must first identify a diameter within the circle. The diameter is a chord that passes through the centre of the circle. Any triangle formed with this diameter as one of its sides, and its third vertex lying on the circumference, will have a right angle at that third vertex. Look for scenarios where a triangle is inscribed within a circle, and one of its sides is explicitly stated or can be inferred to be a diameter. This theorem is a powerful tool for finding unknown angles and proving geometric facts, particularly when solving problems involving constructions within circles.

Applications in Problem Solving

The Angle in a Semicircle Theorem is frequently used in geometry problems involving circles and triangles. It can help in calculating unknown angles, determining if a triangle is right-angled without direct measurement, and proving properties in more complex geometric figures. For example, if you are given a circle with a diameter and a point on the circumference, you can immediately deduce a 90° angle. This can then be used with other angle properties (e.g., angle sum of a triangle, angles on a straight line) or Pythagoras' theorem if side lengths are involved, to solve for further unknowns. It's a key tool in demonstrating that specific lines are perpendicular or that triangles are right-angled.

Understanding the Theorem

The 'Angles in the Same Segment' theorem states that any two angles subtended by the same arc (or chord) at the circumference of a circle are equal. This means if you have a chord AB in a circle, and points C and D are both on the circumference and on the same side of the chord AB, then the angle ∠ACB will always be equal to the angle ∠ADB. This applies regardless of the positions of C and D, as long as they are in the same segment.

Identifying Same Segments

To identify angles in the same segment, look for a common chord (or arc) that subtends both angles. The vertices of the angles must lie on the circumference. Crucially, the vertices must be on the 'same side' of the chord. For instance, if chord PQ is drawn, and points R and S are on the circumference, ∠PRQ and ∠PSQ are equal if R and S are in the same segment defined by PQ. Care must be taken not to confuse this with angles formed by parts of a cyclic quadrilateral on opposite sides of a diagonal.

Applications in Problem Solving

This theorem is a powerful tool for finding unknown angles in circle diagrams. Often, you might be given one angle and need to find another that is in the same segment. It can also be combined with other circle theorems, such as angles in a cyclic quadrilateral (opposite angles sum to 180°), angles subtended at the centre (twice the angle at the circumference), or angles in a semicircle (always 90°). Look for diagrams where a common chord appears to form multiple angles at the circumference.

Converse of the Theorem

The converse of the angles in the same segment theorem is also true and useful: If two angles subtended by a line segment at two points on the same side of the segment are equal, then the four points are concyclic (i.e., they lie on the circumference of a single circle). This can be used to prove that points lie on a circle, which is a common type of geometry problem in higher mathematics.