Prove that any four vertices of a regular pentagon are concylic.

Let the regular pentagon be ABCDE. Since it's regular, and all interior angles are equal to 108°. ∴ ∠ABC = ∠BCD = ∠CDE = ∠DEA = ∠EAB Interior angle of a regular polygon is given by : So, each interior angle of a regular pentagon = - 108°. In triangle AED, AE = ED [Sides of a regular pentagon are equal] ∴ ∠EAD = ∠EDA [Angles opposite to equal sides are equal] ......(1) ⇒ ∠AED + ∠EAD + ∠EDA = 180° ⇒ 108° + ∠EAD + ∠EAD = 180° ⇒ 2∠EAD = 180° - 108° ⇒ 2∠EAD = 72° ⇒ ∠EAD = ⇒ ∠EAD = 36° ∴ ∠EDA = 36°. From figure, ⇒ ∠BAD = ∠BAE - ∠EAD = 108° - 36° = 72°. In quadrilateral ABCD, ∠BAD + ∠BCD = 72° + 108° = 180°. Since, sum of opposite angles = 180°, which is possible when the quadrilateral is cyclic quadrilateral. Hence, proved that any four vertices of a regular pentagon are concylic.

In the figure (ii) given below, AP and BP are tangents to the circle with centre O. If ∠CBP = 25° and ∠CAP = 40°, find
(i) ∠ADB
(ii) ∠AOB
(iii) ∠ACB
(iv) ∠APB.

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In the given figure AC is the diameter of the circle with center O. CD is parallel to BE.
∠AOB = 80° and ∠ACE = 20°. Calculate :
(a) ∠BEC
(b) ∠BCD
(c) ∠CED

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In the given figure (drawn not to scale) chords AD and BC intersect at P, where AB = 9 cm, PB = 3 cm and PD = 2 cm.
(a) Prove that △ APB ~ △ CPD
(b) Find the length of CD
(c) Find area △ APB : area △ CPD.

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In the given figure PT is a tangent to the circle. Chord BA produced meets the tangent PT at P. Given PT = 20 cm and PA = 16 cm.
(a) Prove △ PTB ~ △ PAT
(b) Find the length of AB.

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Mathematics

Prove that any four vertices of a regular pentagon are concyclic.

Circles

Related Questions

In the figure (ii) given below, AP and BP are tangents to the circle with centre O. If ∠CBP = 25° and ∠CAP = 40°, find
(i) ∠ADB
(ii) ∠AOB
(iii) ∠ACB
(iv) ∠APB.

In the given figure AC is the diameter of the circle with center O. CD is parallel to BE.
∠AOB = 80° and ∠ACE = 20°. Calculate :
(a) ∠BEC
(b) ∠BCD
(c) ∠CED

In the given figure (drawn not to scale) chords AD and BC intersect at P, where AB = 9 cm, PB = 3 cm and PD = 2 cm.
(a) Prove that △ APB ~ △ CPD
(b) Find the length of CD
(c) Find area △ APB : area △ CPD.

In the given figure PT is a tangent to the circle. Chord BA produced meets the tangent PT at P. Given PT = 20 cm and PA = 16 cm.
(a) Prove △ PTB ~ △ PAT
(b) Find the length of AB.

Mathematics

Prove that any four vertices of a regular pentagon are concyclic.

Circles

Related Questions

In the figure (ii) given below, AP and BP are tangents to the circle with centre O. If ∠CBP = 25° and ∠CAP = 40°, find(i) ∠ADB(ii) ∠AOB(iii) ∠ACB(iv) ∠APB.

In the given figure AC is the diameter of the circle with center O. CD is parallel to BE.∠AOB = 80° and ∠ACE = 20°. Calculate :(a) ∠BEC(b) ∠BCD(c) ∠CED

In the given figure (drawn not to scale) chords AD and BC intersect at P, where AB = 9 cm, PB = 3 cm and PD = 2 cm.(a) Prove that △ APB ~ △ CPD(b) Find the length of CD(c) Find area △ APB : area △ CPD.

In the given figure PT is a tangent to the circle. Chord BA produced meets the tangent PT at P. Given PT = 20 cm and PA = 16 cm.(a) Prove △ PTB ~ △ PAT(b) Find the length of AB.

In the figure (ii) given below, AP and BP are tangents to the circle with centre O. If ∠CBP = 25° and ∠CAP = 40°, find
(i) ∠ADB
(ii) ∠AOB
(iii) ∠ACB
(iv) ∠APB.

In the given figure AC is the diameter of the circle with center O. CD is parallel to BE.
∠AOB = 80° and ∠ACE = 20°. Calculate :
(a) ∠BEC
(b) ∠BCD
(c) ∠CED

In the given figure (drawn not to scale) chords AD and BC intersect at P, where AB = 9 cm, PB = 3 cm and PD = 2 cm.
(a) Prove that △ APB ~ △ CPD
(b) Find the length of CD
(c) Find area △ APB : area △ CPD.

In the given figure PT is a tangent to the circle. Chord BA produced meets the tangent PT at P. Given PT = 20 cm and PA = 16 cm.
(a) Prove △ PTB ~ △ PAT
(b) Find the length of AB.