Two chords AB and AC of a circle are equal. Prove that the centre of the circle lies on the bisector of angle BAC.

Given: Two chords AB and AC of a circle are equal. The bisector of angle ∠BAC intersects chord BC at point P. To Prove: The center of the circle lies on the bisector of ∠BAC. Construction: Join BC. Proof: In Δ APB and Δ APC, AB = AC (Given) ∠BAP = ∠CAP (Given) AP = AP (Common Side) By using ASA congruency criterion, Δ APB ≅ Δ APC By corresponding parts of congruent triangles, BP = CP and ∠APB = ∠APC ∠APB and ∠APC form a linear pair(they lie on the straight line BC). ⇒ ∠APB + ∠APC = 180° ⇒ ∠APB + ∠APB = 180° ⇒ 2∠APB = 180° ⇒ ∠APB = ⇒ ∠APB = 90° Therefore, AP is the perpendicular to chord BC. The perpendicular bisector of a chord passes through the center of the circle. Hence, the centre of the circle lies on the bisector of ∠BAC.

In the given figure, the diameter CD of a circle with centre O is perpendicular to the chord AB.
If AB = 8 cm and CM = 2 cm, find the radius of the circle.

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Two chords AB and CD of lengths 24 cm and 10 cm respectively of a circle are parallel. If the chords lie on the same side of the centre and distance between them is 7 cm, find the length of a diameter of the circle.

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In the given figure, arc APB : arc BQC = 2 : 3 and ∠AOC = 150°. Find :
(i) ∠AOB
(ii) ∠BOC
(iii) ∠OBA
(iv) ∠OCB
(v) ∠ABC

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In given figure, AB is a side of a regular pentagon and BC is the side of a regular hexagon. Find :
(i) ∠AOB
(ii) ∠OBC

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Mathematics

Two chords AB and AC of a circle are equal. Prove that the centre of the circle lies on the bisector of angle BAC.

Circles

Related Questions

In the given figure, the diameter CD of a circle with centre O is perpendicular to the chord AB.
If AB = 8 cm and CM = 2 cm, find the radius of the circle.

Two chords AB and CD of lengths 24 cm and 10 cm respectively of a circle are parallel. If the chords lie on the same side of the centre and distance between them is 7 cm, find the length of a diameter of the circle.

In the given figure, arc APB : arc BQC = 2 : 3 and ∠AOC = 150°. Find :
(i) ∠AOB
(ii) ∠BOC
(iii) ∠OBA
(iv) ∠OCB
(v) ∠ABC

In given figure, AB is a side of a regular pentagon and BC is the side of a regular hexagon. Find :
(i) ∠AOB
(ii) ∠OBC

Mathematics

Two chords AB and AC of a circle are equal. Prove that the centre of the circle lies on the bisector of angle BAC.

Circles

Related Questions

In the given figure, the diameter CD of a circle with centre O is perpendicular to the chord AB.If AB = 8 cm and CM = 2 cm, find the radius of the circle.

Two chords AB and CD of lengths 24 cm and 10 cm respectively of a circle are parallel. If the chords lie on the same side of the centre and distance between them is 7 cm, find the length of a diameter of the circle.

In the given figure, arc APB : arc BQC = 2 : 3 and ∠AOC = 150°. Find :(i) ∠AOB(ii) ∠BOC(iii) ∠OBA(iv) ∠OCB(v) ∠ABC

In given figure, AB is a side of a regular pentagon and BC is the side of a regular hexagon. Find :(i) ∠AOB(ii) ∠OBC

In the given figure, the diameter CD of a circle with centre O is perpendicular to the chord AB.
If AB = 8 cm and CM = 2 cm, find the radius of the circle.

In the given figure, arc APB : arc BQC = 2 : 3 and ∠AOC = 150°. Find :
(i) ∠AOB
(ii) ∠BOC
(iii) ∠OBA
(iv) ∠OCB
(v) ∠ABC

In given figure, AB is a side of a regular pentagon and BC is the side of a regular hexagon. Find :
(i) ∠AOB
(ii) ∠OBC