In the given diagram, ABCDEF is a regular hexagon inscribed in a circle with centre O. PQ is a tangent to the circle at D. Find the value of :
(a) ∠FAG
(b) ∠BCD
(c) ∠PDE

Question

In the given diagram, ABCDEF is a regular hexagon inscribed in a circle with centre O. PQ is a tangent to the circle at D. Find the value of : (a) ∠FAG (b) ∠BCD (c) ∠PDE

KnowledgeBoat · Accepted Answer

(a) By formula, Each interior angle of a n sided polygon = ∴ ∠FAB = 120° From figure, ∠FAG and ∠FAB form a linear pair. ∴ ∠FAB + ∠FAG = 180° ⇒ 120° + ∠FAG = 180° ⇒ ∠FAG = 180° - 120° = 60°. Hence, ∠FAG = 60°. (ii) Each interior angle of regular hexagon = 120°. Hence, ∠BCD = 120°. (iii) Join FC. We know that, Angle in a semicircle is a right angle. ∴ ∠FBC = 90° In triangle BCD, ⇒ BC = CD (Sides of regular hexagon) ⇒ ∠CBD = ∠CDB = x (Angles opposite to equal sides are equal) By angle sum property of triangle, ⇒ ∠BCD + ∠CDB + ∠CBD = 180° ⇒ 120° + x + x = 180° ⇒ 2x = 180° - 120° ⇒ 2x = 60° ⇒ x = = 30°. From figure, ⇒ ∠FBD = ∠FBC - ∠CBD = 90° - x = 90° - 30° = 60°. We know that, The angle between a tangent and a chord through the point of contact is equal to an angle in the alternate segment. ⇒ ∠PDE = ∠FBD = 60°. Hence, ∠PDE = 60°.

Mathematics

In the given diagram, ABCDEF is a regular hexagon inscribed in a circle with centre O. PQ is a tangent to the circle at D. Find the value of :
(a) ∠FAG
(b) ∠BCD
(c) ∠PDE

Circles

Related Questions

Construct a triangle ABC such that AB = 7 cm, BC = 6 cm and CA = 5 cm.
(a) Draw the locus of the points such that
(i) it is equidistant from BC and BA.
(ii) it is equidistant from points A and B.
(b) Mark P where the loci (i) and (ii) meet, measure and write length of PA.

In the given figure O is the centre of the circle. ABCD is a quadrilateral where sides AB, BC, CD and DA touch the circle at E, F, G and H respectively. If AB = 15 cm, BC = 18 cm and AD = 24 cm, find the length of CD.

AB and CD intersect at the center O of the circle given in the above diagram. If ∠EBA = 33° and ∠EAC = 82°, find
(a) ∠BAE
(b) ∠BOC
(c) ∠ODB

Mathematics

In the given diagram, ABCDEF is a regular hexagon inscribed in a circle with centre O. PQ is a tangent to the circle at D. Find the value of :(a) ∠FAG(b) ∠BCD(c) ∠PDE

Circles

Related Questions

Construct a triangle ABC such that AB = 7 cm, BC = 6 cm and CA = 5 cm.(a) Draw the locus of the points such that(i) it is equidistant from BC and BA.(ii) it is equidistant from points A and B.(b) Mark P where the loci (i) and (ii) meet, measure and write length of PA.

In the given figure O is the centre of the circle. ABCD is a quadrilateral where sides AB, BC, CD and DA touch the circle at E, F, G and H respectively. If AB = 15 cm, BC = 18 cm and AD = 24 cm, find the length of CD.

AB and CD intersect at the center O of the circle given in the above diagram. If ∠EBA = 33° and ∠EAC = 82°, find(a) ∠BAE(b) ∠BOC(c) ∠ODB

In the given diagram, ABCDEF is a regular hexagon inscribed in a circle with centre O. PQ is a tangent to the circle at D. Find the value of :
(a) ∠FAG
(b) ∠BCD
(c) ∠PDE

Construct a triangle ABC such that AB = 7 cm, BC = 6 cm and CA = 5 cm.
(a) Draw the locus of the points such that
(i) it is equidistant from BC and BA.
(ii) it is equidistant from points A and B.
(b) Mark P where the loci (i) and (ii) meet, measure and write length of PA.

AB and CD intersect at the center O of the circle given in the above diagram. If ∠EBA = 33° and ∠EAC = 82°, find
(a) ∠BAE
(b) ∠BOC
(c) ∠ODB