CIRCLES-Notes
Maths - Notes
Angle Subtended by a Chord at a Point
When a chord is drawn inside a circle, it divides the circle into two parts — known as
arcs.
If we take any point on the circumference of the circle (other than the endpoints of the
chord) and join
it to the two ends of the chord, an angle is formed at that point.
This angle is called the angle subtended by the chord at a point on the circle.
Let’s understand this with a simple figure.
Suppose \(AB\) is a chord of a circle with center \(O\).
Take a point \(P\) on the circle (not on the chord). When you join \(P\) to \(A\) and
\(P\) to \(B\), you get an angle \(\angle APB.\)
This angle \(\angle APB\) say \((\theta)\) is called the angle subtended by chord \(AB\)
at
point \(P\) on the circle.
Similarly, if we take another point \(Q\) on the same arc as \(P\), the chord \(AB\)
will subtend another angle
\(\angle AQB \text{ say } (\beta)\) at that point.
Relationship between the size of the chord and the angle subtended by it at the centre.
The larger the chord, the larger is the angle subtended by it at the centre of the circle.
Conversely, the smaller the chord, the smaller is the angle subtended at the centre.
This relationship is perfectly consistent with how the arc behaves:
- A longer chord forms a bigger arc,
- While a shorter chord forms a smaller arc, and the centre angle depends on the arc between the endpoints of the chord.
Theorem-1
Equal chords of a circle subtend equal angles at the centre.
Given:two equal chords AB and CD of a circle with centre O
To Prove:
\[\angle AOB=\angle COD\] Proof:
In \(\triangle AOB\) and \(\triangle COD\)
\[ \begin{aligned} OA&=OC\text{ (Radii of Circle)}\\ OB&=OD \text{ (Radii of Circle)}\\ AB&=CD \text{ (Given)}\\\text{therefore,}\\ \triangle AOB&\cong \triangle COD \text{ (SSS Rule)} \end{aligned}\] \[\implies \boxed{\angle AOB=\angle COD}\text{ (CPCT)}\]Theorem-2
If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.
Given:\[\angle AOB=\angle COD\]
To Prove:
\[AB=CD\] Proof:
In \(\triangle AOB\) and \(\triangle COD\)
\[ \begin{aligned} OA&=OC\text{ (Radii of Circle)}\\ OB&=OD \text{ (Radii of Circle)}\\ \angle AOB&=\angle COD \text{ (Given)}\\\text{therefore,}\\ \triangle AOB&\cong \triangle COD \text{ (SAS Rule)} \end{aligned}\] \[\implies \boxed{AB=CD}\text{ (CPCT)}\]Theorem-3
The perpendicular from the centre of a circle to a chord bisects the chord.
Given:\[\angle OMA=\angle OMB=90^\circ\]
To Prove:
\[MA=MB\] Proof:
In \(\triangle OAM\) and \(\triangle OBM\) \[ \begin{aligned} OA&=OB\text{ (Radii of Circle)}\\ OM&=OM \text{ (Common Side)}\\ \angle OMA&=\angle OMB=90^\circ \text{ (Given)}\\\text{therefore,}\\ \triangle OAM&\cong \triangle OBM \text{ (RHS Rule)} \end{aligned}\] \[\implies \boxed{MA=MB}\text{ (CPCT)}\]
Theorem-4
The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
Given:\[MA=MB\]
To Prove:
\[\angle OMA=\angle OMB=90^\circ\] Proof:
In \(\triangle OAM\) and \(\triangle OBM\)
\[ \begin{aligned} OA&=OB\text{ (Radii of Circle)}\\ OM&=OM \text{ (Common Side)}\\ MA&=MB \text{ (Given)}\\\text{therefore,}\\ \triangle OAM&\cong \triangle OBM \text{ (SSS Rule)} \end{aligned}\] \[\implies \angle OMA=\angle OMB\text{ (CPCT)}\\\\ \angle OMA+\angle OMB=180^\circ\\\text{(Linear Pair)}\\\\ \implies 2\angle OMA=180^\circ\\\\ \implies \boxed{\angle OMA=\angle OMB=90^\circ} \]Theorem-5
Equal chords of a circle (or of congruent circles) are equidistant from the centre (or centres).
Theorem-6
Chords equidistant from the centre of a circle are equal in length.
Theorem-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.
Given:Given an arc \(PQ\) of a circle subtending \(\angle POQ\) at the centre of the Circle \(O\) and \(\angle PAQ\) on a point A on the remaining part of the circle
To Prove:
$$\angle POQ=2\angle PAQ$$ Construction:Join AO and extending to B
Proof:
$$\angle BOQ=\angle OAQ+\angle AQO\tag{1}$$ (Exterior Angle of triangle is equal to sum of the two interior opposite Angles)
similarly,
$$\angle POB=\angle OAP+\angle OPA\tag{2}$$ \[\angle OAQ=\angle AQO\\\text{Equal angles of Isosceles triangles}\\\\ \angle OAP=\angle OPA\\\text{Equal angles of Isosceles triangles}\] Adding equation (1) and (2) $$\scriptsize\begin{aligned} \angle BOQ+\angle POB&=\angle OAQ+\angle AQO+\angle OAP+\angle OPA\\ \angle POQ&=2\angle OAQ+2\angle OAP\\ &=2\left[\angle OAQ+\angle OAP\right] \\ &=2\angle PAQ\\ \angle POQ&=2\angle PAQ\end{aligned}$$ \[\boxed{\angle POQ=2\angle PAQ}\]
Theorem-8
Angles in the same segment of a circle are equal.
Theorem-9
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)
Cyclic Quadrilaterals
A quadrilateral is called cyclic if all its four vertices lie on a single circle.
The circle in which the quadrilateral is drawn is known as its circumcircle.
The sum of the measures of the opposite angles of a cyclic quadrilateral is
always \(180^\circ.\)
\[\angle A+\angle C=180^\circ\\
\angle B+\angle D=180^\circ\]
Theorem-10
The sum of either pair of opposite angles of a cyclic quadrilateral is 180º.
Theorem-11
If the sum of a pair of opposite angles of a quadrilateral is 180º, the quadrilateral is cyclic.
Imporant Points
- A circle is the collection of all points in a plane, which are equidistant from a fixed point in the plane.
- Equal chords of a circle (or of congruent circles) subtend equal angles at the centre.
- If the angles subtended by two chords of a circle (or of congruent circles) at the centre (corresponding centres) are equal, the chords are equal.
- The perpendicular from the centre of a circle to a chord bisects the chord.
- The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
- Equal chords of a circle (or of congruent circles) are equidistant from the centre (or corresponding centres).
- Chords equidistant from the centre (or corresponding centres) of a circle (or of congruent circles) are equal.
- If two arcs of a circle are congruent, then their corresponding chords are equal and conversely if two chords of a circle are equal, then their corresponding arcs (minor, major) are congruent.
- Congruent arcs of a circle subtend equal angles at the centre.
- 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.
- Angles in the same segment of a circle are equal
- Angle in a semicircle is a right angle.
- 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.
- The sum of either pair of opposite angles of a cyclic quadrilateral is \(180^\circ.\)
- If sum of a pair of opposite angles of a quadrilateral is \(180^\circ\), the quadrilateral is cyclic.
