The given figure show a circle with centre O. Also, PQ = QR = RS and ∠PTS = 75°.
Calculate:
(i) ∠POS,
(ii) ∠QOR,
(iii) ∠PQR.

Question

The given figure show a circle with centre O. Also, PQ = QR = RS and ∠PTS = 75°. Calculate: (i) ∠POS, (ii) ∠QOR, (iii) ∠PQR.

KnowledgeBoat · Accepted Answer

Join OP, OQ, OR and OS. Given, PQ = QR = RS ∴ ∠POQ = ∠QOR = ∠ROS = x (let) [Equal chords subtends equal angles at the centre] Arc PQRS subtends ∠POS at the centre and ∠PTS at the circumference of the circle. Thus, ∠POS = 2 x ∠PTS = 2 x 75° = 150° [As angle subtended at the centre by the arc is double that it subtends at any point on the circumference of the circle.] ⇒ ∠POQ + ∠QOR + ∠ROS = 150° ⇒ x + x + x = 150° ⇒ 3x = 150° ⇒ x = = 50°. In ∆OPQ we have, ⇒ OP = OQ [Radii of the same circle] ⇒ ∠OPQ = ∠OQP = y (let) [Angles opposite to equal sides are equal] In ∆OPQ, ⇒ ∠OPQ + ∠OQP + ∠POQ = 180° ⇒ ∠OPQ + ∠OQP + 50° = 180° ⇒ ∠OPQ + ∠OQP = 180° - 50° ⇒ ∠OPQ + ∠OQP = 130° ⇒ 2y = 130° ⇒ y = = 65° ⇒ ∠OPQ = ∠OQP = y = 65°. In ∆OQR we have, ⇒ OQ = OR [Radii of the same circle] ⇒ ∠OQR = ∠ORQ = z (let) [Angles opposite to equal sides are equal] In ∆OQR ⇒ ∠OQR + ∠ORQ + ∠QOR = 180° ⇒ z + z + 50° = 180° ⇒ 2z = 180° - 50° ⇒ 2z = 130° ⇒ z = = 65° ⇒ ∠OQR = ∠ORQ = z = 65°. (i) Hence, ∠POS = 150°. (ii) Hence, ∠QOR = 50°. (iii) From figure, ∠PQR = ∠PQO + ∠OQR = 65° + 65° = 130°. Hence, ∠PQR = 130°.

Mathematics

The given figure show a circle with centre O. Also, PQ = QR = RS and ∠PTS = 75°.
Calculate:
(i) ∠POS,
(ii) ∠QOR,
(iii) ∠PQR.

Circles

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In the following figure, AD is the diameter of the circle with centre O. Chords AB, BC and CD are equal. If ∠DEF = 110°, calculate :
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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 :
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(i) its other two sides are equal.
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Mathematics

The given figure show a circle with centre O. Also, PQ = QR = RS and ∠PTS = 75°.Calculate:(i) ∠POS,(ii) ∠QOR,(iii) ∠PQR.

Circles

Related Questions

In a regular pentagon ABCDE, inscribed in a circle; find the ratio between angle EDA and angle ADC.

In the following figure, AD is the diameter of the circle with centre O. Chords AB, BC and CD are equal. If ∠DEF = 110°, calculate :(i) ∠AEF, (ii) ∠FAB.

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.

If two sides of a cyclic-quadrilateral are parallel; prove that :(i) its other two sides are equal.(ii) its diagonals are equal.

The given figure show a circle with centre O. Also, PQ = QR = RS and ∠PTS = 75°.
Calculate:
(i) ∠POS,
(ii) ∠QOR,
(iii) ∠PQR.

In the following figure, AD is the diameter of the circle with centre O. Chords AB, BC and CD are equal. If ∠DEF = 110°, calculate :
(i) ∠AEF, (ii) ∠FAB.

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.

If two sides of a cyclic-quadrilateral are parallel; prove that :
(i) its other two sides are equal.
(ii) its diagonals are equal.