Notes On Arcs of a Circle - CBSE Class 9 Maths

A part of a circle is called an arc. Arcs of a circle that superimpose each other completely are called congruent arcs. If two arcs of a circle are congruent, then their corresponding chords are equal. Conversely, if two chords of a circle are equal, then their corresponding arcs are congruent.

chords, equal chords, arcs, equal arcs, corresponding arcs, corresponding chords, congruent arcs, congruent chords, arcs of a circle

Corresponding arcs of two equal chords of a circle are congruent.

Congruent arcs of a circle subtend equal angles at the centre.
                                      
arcs, congruent arcs, chords, angles at centre, equal angles at centre

Given: Two congruent arcs AB and CD.
To prove: ∠ AOB = ∠COD
Construction: Draw chords AB and CD.
Proof: The angle subtended by an arc at the centre is equal to the angle subtended by its corresponding chord at the centre.
In the given figure,
AB = CD (Chords corresponding to congruent arcs of a circle are equal)
∠AOB = ∠COD (Equal chords subtend equal angles at the centre)
Hence, the theorem is proved.

arcs, congruent arcs, chords, angles at centre, equal angles at centre

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



Given: Arc AB. Point C on the circle is outside AB.
To prove: ∠AOB = 2 × ∠ACB
Construction: Draw a line CO extended till point D.
Proof: In ΔOAC in each of these figures,
∠AOD = ∠OAC + ∠OCA (Exterior angle of a triangle is equal to sum of two opposite interior angles)
OA = OC (Radii of same circle)
Thus, ∠ OAC = ∠ OCA (Angles opposite equal sides of a triangle are equal)
∠AOD = ∠OAC + ∠OCA
⇒∠AOD = 2 × ∠OCA
Similarly, in ΔOBC, ∠BOD = 2 × ∠OCB
∠AOD = 2 × ∠OCA and ∠BOD = 2 × ∠OCB
⇒ ∠AOD + ∠BOD = 2 ∠OCA + 2 ∠OCB
∠AOD + ∠BOD = 2 × (∠OCA + ∠OCB)
or ∠AOB = 2 × ∠ACB
Hence, the theorem is proved.

Angles subtended by an arc at all points within the same segment of the circle are equal.

angles subtended by arc, same segment, segment, chords

Given: An arc AB. Points C and D are on the circle in the same segment.
To prove: ∠ACB = ∠ADB
Proof: By the theorem that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the circle:
∠AOB = 2 × ∠ACB
Also, ∠AOB = 2 × ∠ADB
∴ ∠ACB = ∠ADB
Hence, the theorem is proved.

All angles formed in a semi circle are right angles.

semi circle, angle in semi-circle, diameter, chord
Given: A circle with centre O, and Q, P and R are three points on the circumference of the circle.
Construction: Join the points Q, P and R to the points A and B.
To prove: ∠AQB = ∠APB = ∠ARB
Proof: ∠AQB = 1 2 ∠AOB [Angle subtended by an arc at the centre is double the angle subtended by the arc at any point on the remaining part of the circle.]
∠APB = 1 2 ∠AOB [Angle subtended by an arc at the centre is double the angle subtended by the arc at any point on the remaining part of the circle.]
Similarly, ∠ARB = 1 2 ∠AOB.

∴ ∠AQB = ∠APB = ∠ARB
Hence, the theorem is proved.

Summary

A part of a circle is called an arc. Arcs of a circle that superimpose each other completely are called congruent arcs. If two arcs of a circle are congruent, then their corresponding chords are equal. Conversely, if two chords of a circle are equal, then their corresponding arcs are congruent.

chords, equal chords, arcs, equal arcs, corresponding arcs, corresponding chords, congruent arcs, congruent chords, arcs of a circle

Corresponding arcs of two equal chords of a circle are congruent.

Congruent arcs of a circle subtend equal angles at the centre.
                                      
arcs, congruent arcs, chords, angles at centre, equal angles at centre

Given: Two congruent arcs AB and CD.
To prove: ∠ AOB = ∠COD
Construction: Draw chords AB and CD.
Proof: The angle subtended by an arc at the centre is equal to the angle subtended by its corresponding chord at the centre.
In the given figure,
AB = CD (Chords corresponding to congruent arcs of a circle are equal)
∠AOB = ∠COD (Equal chords subtend equal angles at the centre)
Hence, the theorem is proved.

arcs, congruent arcs, chords, angles at centre, equal angles at centre

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



Given: Arc AB. Point C on the circle is outside AB.
To prove: ∠AOB = 2 × ∠ACB
Construction: Draw a line CO extended till point D.
Proof: In ΔOAC in each of these figures,
∠AOD = ∠OAC + ∠OCA (Exterior angle of a triangle is equal to sum of two opposite interior angles)
OA = OC (Radii of same circle)
Thus, ∠ OAC = ∠ OCA (Angles opposite equal sides of a triangle are equal)
∠AOD = ∠OAC + ∠OCA
⇒∠AOD = 2 × ∠OCA
Similarly, in ΔOBC, ∠BOD = 2 × ∠OCB
∠AOD = 2 × ∠OCA and ∠BOD = 2 × ∠OCB
⇒ ∠AOD + ∠BOD = 2 ∠OCA + 2 ∠OCB
∠AOD + ∠BOD = 2 × (∠OCA + ∠OCB)
or ∠AOB = 2 × ∠ACB
Hence, the theorem is proved.

Angles subtended by an arc at all points within the same segment of the circle are equal.

angles subtended by arc, same segment, segment, chords

Given: An arc AB. Points C and D are on the circle in the same segment.
To prove: ∠ACB = ∠ADB
Proof: By the theorem that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the circle:
∠AOB = 2 × ∠ACB
Also, ∠AOB = 2 × ∠ADB
∴ ∠ACB = ∠ADB
Hence, the theorem is proved.

All angles formed in a semi circle are right angles.

semi circle, angle in semi-circle, diameter, chord
Given: A circle with centre O, and Q, P and R are three points on the circumference of the circle.
Construction: Join the points Q, P and R to the points A and B.
To prove: ∠AQB = ∠APB = ∠ARB
Proof: ∠AQB = 1 2 ∠AOB [Angle subtended by an arc at the centre is double the angle subtended by the arc at any point on the remaining part of the circle.]
∠APB = 1 2 ∠AOB [Angle subtended by an arc at the centre is double the angle subtended by the arc at any point on the remaining part of the circle.]
Similarly, ∠ARB = 1 2 ∠AOB.

∴ ∠AQB = ∠APB = ∠ARB
Hence, the theorem is proved.

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