Arc of a circle:
The portion between any two points on the circumference of a circle is called an arc.
Important!
- The arc connecting the two points on the circle along the circle's edge are called end points.
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If the ends of the arc coincides with the ends of the diameter in such a case each arc is called a semicircle.
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The area enclosed by a semicircle and the diameter of a circle is called the semicircular region.
Segment of a circle:
The region enclosed by the chord and the arc is called the segment of the circle.
Important!
The region enclosed by the chord and the minor arc is called the minor segment, and the region enclosed by the chord and the major arc is called the major segment.
Angles Subtended by an Arc:
The angle subtended by the arc \(AB\) at the centre is the measure of the \(\angle AOB\), as we sweep along the arc.
- Here the arc through \(X\) subtends an \(\angle AOB\) is the angle subtended by the minor arc.
- And, the arc through \(Y\) subtends an \(\angle AOB\) is the angle subtended by the major arc.
Important!
The angle subtended by the major arc at the centre of the circle is also called a reflex angle and is greater than \(180^{\circ}\).
Theorem on angles subtended by an arc:
The angle subtended by an arc at the centre of the circle is double the angle subtended by the arc at any point on the circle outside the arc.
Special cases of angle at the centre and the circumference:
Conjecture \(1\)
The angle subtended by a diameter at any point on the circle is \(90°\).
Explanation:
The conjecture is that any angle inscribed in the semi-circle is always \(90^{\circ}\). In other words, the angle subtended by the diameter is always \(90^{\circ}\).
Conjecture \(2\)
Theorem: Angles in the same segment of a circle are equal.
Explanation:
The theorem states that angles in the same segment of the circle are equal. In the given circle, the angles \(\angle PRQ\) and \(\angle PSQ\) are equal as they lie on the same segment (i.e.) \(\angle PRQ = \angle PSQ\).
Cyclic Quadrilateral:
A four-sided figure inscribed in a circle such that all its vertices lie on the circumference of the circle is said to be a cyclic quadrilateral.
Important!
- A quadrilateral are also called a \(4\)-gon.
- The angle subtended by the major arc at the centre \(O\) is called the reflex angle
Theorem on a cyclic quadrilateral:
The sum of two opposite angles of a cyclic quadrilateral is \(180^{\circ}\).
Converse of Theorem: If two opposite angles of a quadrilateral add up to \(180°\), then the vertices of the quadrilateral lie on a circle, i.e., they are concyclic.
Important!
The exterior angle of a cyclic quadrilateral at a vertex is equal to the interior opposite angle.
Theorem: Concyclic
If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle (i.e. they are concyclic).