Question

A,B,C are points on the circle with centre O. If arc BC =110, m(arc AB) =125, find m(arc ABC) and m(arc AC)​

Answer 1

Answer:

115

Step-by-step explanation:

Take any point P on the circumference of the circle as shown.

Join AP and CP.

∵ABC subtends ∠AOC at centre O and ∠APC at any point P on the circumference of the circle.

∴∠AOC=2∠APC ...[∵ Angle subtended by an arc at the centre is twice the angle at the circumference]

∴∠APC=  

2

1

​  

∠AOC[∵AOC=130  

∘

]

⟹  

2

1

​  

×130  

∘

=65  

∘

.

∵ ABCP is a cycle quadrilateral,

⟹∠APC+∠ABC=180  

∘

 ...[∵ sum of opposite angles of a cyclic quadrilateral is 180  

∘

]

⟹65  

∘

+∠ABC=180  

∘

 

∴∠ABC=180  

∘

−65  

∘

=115  

∘

.

Hence, option B is correct.

Answered By

Divyanshu

Answer 2

In a circle angle subtended by any chord on opposite sides are supplementary.

So m(arcAPC)+m(arcABC)=180

o

So m(arcABC)=180

o

−60

o

=120

o