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In the given figure, AB is a side of a regular six-sided polygon and AC is a side of a regular eight sided polygon inscribed in the circle with centre O. Calculate the sizes of:

(i) ∠AOB, (ii) ∠ACB (iii) ∠ABC

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#### Solution

(i) Arc AB subtends ∠AOB at the centre and

∠ACB at the remaining part of the circle.

∴ `∠ACB = 1/2∠AOB`

Since AB is the side of a regular hexagon,

∠AOB = 60°

(ii) `∠AOB = 60° ⇒ ∠ACB = 1/2 xx60° = 30°`

(iii) Since AC is the side of a regulare octagon,

`∠AOC = (360°) /8= 45°`

Again, Arc AC subtends ∠AOC at the center and

∠ABC at the remaining part of the circle.

⇒ `∠ABC = 1/2 ∠AOC`

⇒ `∠ABC = (45°)/2 = 22.5°`

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