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
(i) ∠AOB, (ii) ∠ACB (iii) ∠ABC
Exercise 17 (B) | Q 5 | Page 265
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(i) Arc AB subtends ∠AOB at the centre and
∠ACB at the remaining part of the circle.
$\therefore \angle A C B=\frac{1}{2} \angle A O B$
Since AB is the side of a regular hexagon,
∠AOB = 60°
(ii) $\angle A O B=60^{\circ} \Rightarrow \angle A C B=\frac{1}{2} \times 60^{\circ}=30^{\circ}$
(iii) Since AC is the side of a regulare octagon,
$\angle A O C=\frac{360^{\circ}}{8}=45^{\circ}$
Again, Arc AC subtends ∠AOC at the center and
∠ABC at the remaining part of the circle.
$\Rightarrow \angle A B C=\frac{1}{2} \angle A O C$
$\Rightarrow \angle A B C=\frac{45^{\circ}}{2}=22.5^{\circ}$
∠ACB at the remaining part of the circle.
$\therefore \angle A C B=\frac{1}{2} \angle A O B$
Since AB is the side of a regular hexagon,
∠AOB = 60°
(ii) $\angle A O B=60^{\circ} \Rightarrow \angle A C B=\frac{1}{2} \times 60^{\circ}=30^{\circ}$
(iii) Since AC is the side of a regulare octagon,
$\angle A O C=\frac{360^{\circ}}{8}=45^{\circ}$
Again, Arc AC subtends ∠AOC at the center and
∠ABC at the remaining part of the circle.
$\Rightarrow \angle A B C=\frac{1}{2} \angle A O C$
$\Rightarrow \angle A B C=\frac{45^{\circ}}{2}=22.5^{\circ}$
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