In the given figure, BD is a side of regular hexagon, DC is a side of a regular pentagon and AD is a diameter. Calculate :
(i) ∠ADC,
(ii) ∠BDA,
(iii) ∠ABC,
(iv) ∠AEC.

Question

In the given figure, BD is a side of regular hexagon, DC is a side of a regular pentagon and AD is a diameter. Calculate : (i) ∠ADC, (ii) ∠BDA, (iii) ∠ABC, (iv) ∠AEC.

KnowledgeBoat · Accepted Answer

Let O be the center of the circle. Join BC, BO, CO and EO. Since, BD is the side of a regular hexagon, ∴ ∠BOD = = 60°. Since, DC is the side of a regular pentagon, ∴ ∠COD = = 72°. In △BOD, OB = OD [Radii of same circle] ∴ ∠OBD = ∠ODB = x (let) [Angles opposite to equal sides are equal] ⇒ ∠OBD + ∠ODB + ∠BOD = 180° ⇒ x + x + 60° = 180° ⇒ 2x = 180° - 60° ⇒ 2x = 120° ⇒ x = = 60°. (i) In △OCD, ⇒ OD = OC [Radii of same circle] ⇒ ∠ODC = ∠OCD = y (let) [Angles opposite to equal sides are equal] ⇒ ∠OCD + ∠ODC + ∠COD = 180° ⇒ y + y + 72° = 180° ⇒ 2y = 180° - 72° ⇒ 2y = 108° ⇒ y = = 54°. From figure, ∠ADC = ∠ODC = y = 54°. Hence, ∠ADC = 54°. (ii) From figure, ∠BDA = ∠BDO = 60°. Hence, ∠BDA = 60°. (iii) We know that, Angle at the centre is twice the angle at remaining circumference. Arc AC subtends ∠AOC at the centre and ∠ABC at the remaining part of the circle. ∴ ∠AOC = 2∠ABC ⇒ ∠ABC = ∠AOC ⇒ ∠ABC = [∠AOD - ∠COD] ⇒ ∠ABC = [180° - 72°] ⇒ ∠ABC = ⇒ ∠ABC = 54°. Hence, ∠ABC = 54°. (iv) In cyclic quadrilateral AECD, ⇒ ∠AEC + ∠ADC = 180° ⇒ ∠AEC + 54° = 180° ⇒ ∠AEC = 180° - 54° ⇒ ∠AEC = 126°. Hence, ∠AEC = 126°.

Mathematics

In the given figure, BD is a side of regular hexagon, DC is a side of a regular pentagon and AD is a diameter. Calculate :
(i) ∠ADC,
(ii) ∠BDA,
(iii) ∠ABC,
(iv) ∠AEC.

Circles

Related Questions

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

The figure shows a circle with centre O. AB is the side of regular pentagon and AC is the side of regular hexagon.
Find the angles of triangle ABC.

In the figure, O is the centre of the circle and the length of arc AB is twice the length of arc BC. If angle AOB = 108°, find :
(i) ∠CAB,
(ii) ∠ADB.

In the given figure, AB = BC = CD and ∠ABC = 132°. Calculate :
(i) ∠AEB,
(ii) ∠AED,
(iii) ∠COD.

Mathematics

In the given figure, BD is a side of regular hexagon, DC is a side of a regular pentagon and AD is a diameter. Calculate :(i) ∠ADC,(ii) ∠BDA,(iii) ∠ABC,(iv) ∠AEC.

Circles

Related Questions

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

The figure shows a circle with centre O. AB is the side of regular pentagon and AC is the side of regular hexagon.Find the angles of triangle ABC.

In the figure, O is the centre of the circle and the length of arc AB is twice the length of arc BC. If angle AOB = 108°, find :(i) ∠CAB,(ii) ∠ADB.

In the given figure, AB = BC = CD and ∠ABC = 132°. Calculate :(i) ∠AEB,(ii) ∠AED,(iii) ∠COD.

In the given figure, BD is a side of regular hexagon, DC is a side of a regular pentagon and AD is a diameter. Calculate :
(i) ∠ADC,
(ii) ∠BDA,
(iii) ∠ABC,
(iv) ∠AEC.

The figure shows a circle with centre O. AB is the side of regular pentagon and AC is the side of regular hexagon.
Find the angles of triangle ABC.

In the figure, O is the centre of the circle and the length of arc AB is twice the length of arc BC. If angle AOB = 108°, find :
(i) ∠CAB,
(ii) ∠ADB.

In the given figure, AB = BC = CD and ∠ABC = 132°. Calculate :
(i) ∠AEB,
(ii) ∠AED,
(iii) ∠COD.