U37 Basic Properties of Circles - Essential Formulas

This section covers the fundamental geometric properties of circles, including relationships between chords, arcs, angles, and tangents. These properties are crucial for solving DSE geometry problems.

1 Angles at Centre and Circumference

Angle at Centre Theorem

The angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at any point on the remaining part of the circumference.

$$ \angle AOB = 2 \times \angle ACB $$
A B C O θ 2θ

2 Angles in the Same Segment

Angles in the Same Segment Theorem

Angles in the same segment of a circle are equal. That is, for points $A$, $B$, $C$, and $D$ on the circumference, if $C$ and $D$ lie on the same arc $AB$, then $\angle ACB = \angle ADB$.

$$ \angle ACB = \angle ADB $$
A B C D O θ θ

3 Cyclic Quadrilaterals

Opposite Angles of a Cyclic Quadrilateral

The opposite angles of a cyclic quadrilateral are supplementary (add up to $180^\circ$). For a cyclic quadrilateral $ABCD$, $\angle ABC + \angle ADC = 180^\circ$ and $\angle BAD + \angle BCD = 180^\circ$.

$$ \angle ABC + \angle ADC = 180^\circ $$
A B C D O θ 180°-θ

4 Tangent Properties

Tangent-Radius Property

A tangent to a circle is perpendicular to the radius drawn to the point of tangency. If $TA$ is a tangent at point $A$ and $O$ is the centre, then $OA \perp TA$.

$$ OA \perp TA $$
O A T

Tangent-Chord Angle Theorem (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. If $TA$ is a tangent at $A$ and $AB$ is a chord, then $\angle TAB = \angle ACB$.

$$ \angle TAB = \angle ACB $$
O A B C T θ θ

Struggling with complex problems?

Learner App features AI step-by-step analysis technology. Snap a photo and it will guide you through the solution!

Download Learner Now