Geometry Revision Notes

The Perpendicular Relationship

In a circle, a radius drawn to the midpoint of a chord is always perpendicular to that chord. This creates a 90-degree angle at the point where the radius intersects the chord. This property is crucial for solving problems involving angles and lengths within circles. It also works in reverse: if a radius is perpendicular to a chord, it must bisect the chord.

Forming Right-Angled Triangles

When a radius meets a chord, especially if the radius is drawn to the midpoint of the chord, a right-angled triangle can be formed. The sides of this triangle are typically the radius, half of the chord, and the segment of the radius from the center to the chord's midpoint. This allows the application of Pythagoras' Theorem ($a^2 + b^2 = c^2$) to find unknown lengths and basic trigonometry (SOH CAH TOA) to find unknown angles.

Using Pythagoras' Theorem

If you know two sides of the right-angled triangle formed by the radius, half-chord, and the segment connecting the center to the chord's midpoint, you can find the third side using Pythagoras' Theorem. For example, if the radius (hypotenuse) and half-chord are known, you can find the distance from the center to the chord. Knowing this distance and the half-chord allows you to find the radius or vice versa.

Applying Trigonometric Ratios

Once a right-angled triangle is identified, trigonometric ratios can be used to find unknown angles or side lengths. For example, if you know the radius and the length of a chord, you can first find half the chord. Then, in the right-angled triangle, you can use the sine, cosine, or tangent functions involving the angle between the radius and the half-chord to determine its value. Remember SOH (Sine = Opposite/Hypotenuse), CAH (Cosine = Adjacent/Hypotenuse), TOA (Tangent = Opposite/Adjacent).

The Fundamental Theorem: Radius Perpendicular to Tangent

A key property in circle geometry is that the angle between a radius and a tangent, at the point where the tangent touches the circle (the point of tangency), is always 90 degrees. This creates a right angle, which is essential for many calculations. This theorem is an absolute rule for all circles and all tangents. When you see a tangent touching a circle, immediately identify the radius drawn to that point as forming a right angle with the tangent.

Applying the 90-degree Property in Problem Solving

Recognising the 90-degree angle between a radius and a tangent is often the first step in solving more complex geometry problems. This right angle can form part of a right-angled triangle, allowing you to use Pythagoras' Theorem to find unknown side lengths or trigonometry (SOH CAH TOA) to find unknown angles. It can also be used in conjunction with other circle theorems, such as angles at the centre and circumference, or properties of quadrilaterals and isosceles triangles within the circle, to solve for various unknown values.

Common Geometric Setups

Typical problems involve tangents drawn from an external point to a circle. In such scenarios, the two tangents from the external point to the circle are equal in length. If radii are drawn to the points of tangency, these radii will each be perpendicular to their respective tangents, creating two right-angled triangles. Often, a quadrilateral is formed by the centre, the external point, and the two points of tangency. The sum of angles in this quadrilateral is 360 degrees, and with two angles known to be 90 degrees, this simplifies calculations significantly.

Understanding the Theorem's Core Statement

The Alternate Segment Theorem states that the angle between a tangent to a circle and a chord drawn from the point of contact is equal to the angle in the alternate segment. This means if you have a tangent line touching a circle at point A, and a chord AB is drawn from A, then the angle formed by the tangent and the chord (e.g., angle TAB) will be equal to any angle subtended by the chord AB in the *opposite* segment of the circle. This is a fundamental concept for solving angle problems in circles involving tangents.

Identifying the Alternate Segment

When a chord is drawn from the point of tangency, it divides the circle into two segments. The 'alternate segment' is the segment that does not contain the angle formed by the tangent and the chord. For example, if chord AB is drawn from the point of tangency A, and the angle between the tangent and the chord is at A, then the alternate segment is the one on the 'other side' of the chord AB. Any angle subtended by chord AB at the circumference in this alternate segment will be equal to the tangent-chord angle.

Applying the Theorem in Problem Solving

To apply the Alternate Segment Theorem, first, identify the tangent to the circle and the point of contact. Next, identify a chord originating from this point of contact. The angle between this chord and the tangent is key. Then, find the angle subtended by this *same chord* in the alternate segment of the circle. These two angles will be equal. This theorem often works in conjunction with other circle theorems, such as 'angles in the same segment are equal' or 'angle in a semicircle is a right angle', to solve more complex problems.

Definition and Basic Property

A cyclic quadrilateral is a quadrilateral whose vertices all lie on the circumference of a circle. The most fundamental property is that opposite angles in a cyclic quadrilateral sum to 180°. For any cyclic quadrilateral ABCD, this means ∠A + ∠C = 180° and ∠B + ∠D = 180°. This property is frequently used to find unknown angles and prove other geometric relationships within circles. It's a cornerstone for more complex problems.

Exterior Angle Property

The exterior angle of a cyclic quadrilateral is equal to its interior opposite angle. If a side of a cyclic quadrilateral is extended, forming an exterior angle, that angle will be equal in measure to the interior angle opposite to the interior angle adjacent to it. For example, if side AB of cyclic quadrilateral ABCD is extended to point E, then the exterior angle ∠EBC = ∠ADC. This property stems directly from the opposite angles summing to 180° rule and supplementary angles on a straight line.

Combining with Other Circle Theorems

Solving complex problems involving cyclic quadrilaterals often requires the application of other circle theorems. These include: angles in the same segment are equal; the angle at the centre is twice the angle at the circumference; the angle in a semicircle is a right angle (90°); and the alternate segment theorem. The alternate segment theorem states that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. Mastery of all these theorems and knowing when and how to combine them is key to higher-level reasoning.

Proof of Cyclic Quadrilateral Properties

Often, problems require proving that a given quadrilateral is cyclic or proving why a particular angle property holds. To prove a quadrilateral is cyclic, you could demonstrate that its opposite angles sum to 180°, or that an exterior angle equals the interior opposite angle. This might involve algebraic manipulation or using other angle properties within the diagram. Sometimes, identifying that all four vertices lie on a circle (e.g., they are equidistant from a central point) is sufficient. These proofs build a deeper understanding of the geometric relationships.