In the following figure, PQ and PR are tangents to the circle, with center O. If ∠QPR = 60°, calculate :
(i) ∠QOR,
(ii) ∠OQR,
(iii) ∠QSR.

Question

In the following figure, PQ and PR are tangents to the circle, with center O. If ∠QPR = 60°, calculate : (i) ∠QOR, (ii) ∠OQR, (iii) ∠QSR.

KnowledgeBoat · Accepted Answer

(i) We know that, The tangent at any point of a circle and the radius through this point are perpendicular to each other. In quadrilateral ORPQ, ∠OQP = ∠ORP = 90° [∵ The tangent at any point of a circle and the radius through this point are perpendicular to each other] ∠QPR = 60° [Given] ⇒ ∠OQP + ∠ORP + ∠QPR + ∠QOR = 360° [By angle sum property of quadrilateral] ⇒ 90° + 90° + 60° + ∠QOR = 360° ⇒ 240° + ∠QOR = 360° ⇒ ∠QOR = 120°. Hence, ∠QOR = 120°. (ii) Join QR. In △QOR, OQ = OR (Radii of same circle) As, angles opposite to equal sides are equal. ∴ ∠OQR = ∠ORQ ..........(1) ⇒ ∠OQR + ∠ORQ + ∠QOR = 180° ⇒ ∠OQR + ∠ORQ + 120° = 180° ⇒ ∠OQR + ∠ORQ = 180° - 120° ⇒ ∠OQR + ∠ORQ = 60° ⇒ ∠OQR + ∠OQR = 60° ⇒ 2∠OQR = 60° ⇒ ∠OQR = ⇒ ∠OQR = 30°. Hence, ∠OQR = 30°. (iii) Arc RQ subtends ∠QOR at the center and ∠QSR at the remaining part of the circle. We know that, The angle subtended by an arc at the centre is twice the angle subtended at the circumference. ⇒ ∠QOR = 2∠QSR ⇒ ∠QSR = ∠QOR = = 60°. Hence, ∠QSR = 60°.

Mathematics

In the following figure, PQ and PR are tangents to the circle, with center O. If ∠QPR = 60°, calculate :
(i) ∠QOR,
(ii) ∠OQR,
(iii) ∠QSR.

Circles

Related Questions

In the given figure, AB is the diameter of the circle, with center O, and AT is the tangent. Calculate the numerical value of x.

ABC is a right angled triangle with AB = 12 cm and AC = 13 cm. A circle, with center O, has been inscribed inside the triangle. Calculate the value of x, the radius of the inscribed circle.

In the given figure, PT touches the circle with center O at point R. Diameter SQ is produced to meet the tangent TR at P.
Given ∠SPR = x° and ∠QRP = y°;
prove that :
(i) ∠ORS = y°
(ii) write an expression connecting x and y.

In a triangle ABC, the incircle (center O) touches BC, CA and AB at points P, Q and R respectively. Calculate :
(i) ∠QOR
(ii) ∠QPR;
given that ∠A = 60°.

Mathematics

In the following figure, PQ and PR are tangents to the circle, with center O. If ∠QPR = 60°, calculate :(i) ∠QOR,(ii) ∠OQR,(iii) ∠QSR.

Circles

Related Questions

In the given figure, AB is the diameter of the circle, with center O, and AT is the tangent. Calculate the numerical value of x.

ABC is a right angled triangle with AB = 12 cm and AC = 13 cm. A circle, with center O, has been inscribed inside the triangle. Calculate the value of x, the radius of the inscribed circle.

In the given figure, PT touches the circle with center O at point R. Diameter SQ is produced to meet the tangent TR at P.Given ∠SPR = x° and ∠QRP = y°;prove that :(i) ∠ORS = y°(ii) write an expression connecting x and y.

In a triangle ABC, the incircle (center O) touches BC, CA and AB at points P, Q and R respectively. Calculate :(i) ∠QOR(ii) ∠QPR;given that ∠A = 60°.

In the following figure, PQ and PR are tangents to the circle, with center O. If ∠QPR = 60°, calculate :
(i) ∠QOR,
(ii) ∠OQR,
(iii) ∠QSR.

In the given figure, PT touches the circle with center O at point R. Diameter SQ is produced to meet the tangent TR at P.
Given ∠SPR = x° and ∠QRP = y°;
prove that :
(i) ∠ORS = y°
(ii) write an expression connecting x and y.

In a triangle ABC, the incircle (center O) touches BC, CA and AB at points P, Q and R respectively. Calculate :
(i) ∠QOR
(ii) ∠QPR;
given that ∠A = 60°.