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In the given circle, arc APB and arc BQC are in the ratio 2 : 5 and O is centre of the circle.

In the given circle, arc APB and arc BQC are in the ratio 2 : 5 and O is centre of the circle. Chapterwise Revision (Stage 2), Concise Mathematics Solutions ICSE Class 9.

If angle AOB = 44°; find angle AOC.

Circles

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Answer

Given, arc APBarc BQC=25\dfrac{\text{arc APB}}{\text{arc BQC}} = \dfrac{2}{5}

Let angle AOC be θ°.

Since the length of an arc is proportional to the central angle subtended by the arc, we can write:

⇒arc APBarc BQC=∠AOB∠BOC⇒44°θ°=25⇒θ°=44°×52⇒θ°=220°2=110°\Rightarrow\dfrac{\text{arc APB}}{\text{arc BQC}} = \dfrac{∠AOB}{∠BOC}\\[1em] \Rightarrow \dfrac{44°}{ θ°} = \dfrac{2}{5}\\[1em] \Rightarrow θ° = \dfrac{44° \times 5}{2}\\[1em] \Rightarrow θ° = \dfrac{220°}{2} = 110°

Now, ∠AOC = ∠AOB + ∠BOC

= 110° + 44° = 154°

Hence, the angle AOC = 154°.

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