Angle Subtended by an Arc of a Circle

⚖️ Equal Chords

In a circle, chords that are equidistant from the centre are equal in length. This fundamental property forms the basis for many circle theorems.

Key Properties:

• Equal chords subtend equal angles at the centre
• Equal chords are equidistant from the centre
• The perpendicular from the centre to a chord bisects the chord

🔄 Arc Superimposition

When two arcs of equal measure are superimposed, they coincide perfectly. This concept is crucial for understanding angle relationships in circles.

Applications:

• Proving equal angles at the centre
• Demonstrating arc congruence
• Understanding central angle properties

⭕ Angle at Centre

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 circumference.

Theorem 9.7:

The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

⭕ Semicircle Angle

Any angle inscribed in a semicircle is a right angle. This is a special case of the angle at centre theorem.

Theorem 9.8:

The angle in a semicircle is a right angle.

Converse:

If an angle inscribed in a circle is a right angle, then the chord subtending it is a diameter.

⬜ Cyclic Quadrilateral

A quadrilateral is called cyclic if all its vertices lie on a circle. Cyclic quadrilaterals have special angle properties.

Theorem 9.10:

Opposite angles of a cyclic quadrilateral are supplementary (add up to 180°).

Theorem 9.11:

The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

💡 Practical Applications

Circle theorems have numerous real-world applications in engineering, architecture, and design.

Real-world Uses:

• Architectural design and construction
• Engineering calculations for circular structures
• Navigation and GPS systems
• Art and design principles
• Mechanical engineering and gear design