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Angle Subtended by an Arc of a Circle on the Circle and at the Center

The given fig...

Question

The given figure shown a circle with centre O. Also, PQ = QR = RS and $∠$ PTS = $75^{o}$ . Calculate :

(i) $∠$ POS,

(ii) $∠$ QOR,

(iii) $∠$ PQR.

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Solution

Given $PQ = QR = RS$

$\Rightarrow \angle POQ = \angle QOR = \angle ROS$ (Equal chords subtend equal angles at the center of a circle)

$\angle POS = 2 \ \angle PTS$ (angle subtended at the center is twice that of the one subtended on the circumference)

$\angle POQ = \angle QOR = \angle ROS = \frac{150}{3} = 50^o$

In $\triangle OPQ$ ,

$OP = OQ$ (radius of the same circle)

$\angle OPQ = \angle OQP$

Therefore $\angle OPQ + \angle OQP + \angle POQ = 180^o$

$\angle OPQ + \angle OQP = 180-50=130^o$

$2 \angle OPQ = 130$

$\angle OPQ = 65^o$

Similarly, in $\triangle OQR, \angle OQR = \angle ORQ = 65^o$

and in $\triangle ORS, \angle ORS = \angle OSR = 65^o$

Therefore

(i) $\angle POS = 150^o$

(ii) $\angle QOR = 50^o$

(iii) $\angle PQR = \angle PQO + \angle OQR = 65+65 = 130^o$

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Similar questions

Q. The given figure shows a circle with centre

O

P Q = Q R = R S

∠ P T S = 75^{o}

. Calculate:
i)

∠ P O S

∠ Q O R

∠ P Q R

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In the following figure, PQ and PR are tangents to the circle, with centre O. If $∠ Q P R = 60^{o}$ , calculate :

(i) $∠ Q O R,$
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Q. In the given figure, chords

P Q = R S, ∠ P O Q = 75^{\circ}

∠ P O S = 150^{\circ}

Find the value of

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The given figure shown a circle with centre O and $∠$ ABP = $42^{o}$ .

Calculate the measure of :

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Q. In the given figure, O is the centre of a circle, PQ is a chord and the tangent PT at P makes an angle of 50° with PQ. Then, ∠POQ = ?

(a) 130°
(b) 100°
(c) 90°
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