Name: Proof of Angle in the Same Segment of a Circle
Uploaded: 2019-07-19
Duration: 1 min 26 s
Description: The most important and simple theorem from the chapter circle that angle in the same segment of the circle are equal. Let's derive this theorem.

Question

Each side of a regular hexagon, inscribed in a circle subtends an angle of

60^{\circ}

at the centre and is equals to the radius of the circle. Prove it.

Accepted Answer

To prove:
Each side of a regular hexagon, inscribed in a circle subtends an angle of $60 °$ at the Centre and is equal to radius of the circle.

Proof:

Since regular hexagon is inscribed in a circle, the sides of the hexagon will be chord of the circle.

So, Chords $AB = BC = CD = DE = EF = FA$

We know that,

In a circle, equal chords subtends equal angles at the Centre.

$∠ AOB = ∠ BOC = ∠ COD = ∠ DOE = ∠ EOF = ∠ AOF = x$ (say)

$\Rightarrow x + x + x + x + x + x = 360 °$ (sum of all angles at the centre is $360 °$

$\Rightarrow 6 x = 360 °$

or $x = 60 °$

Hence, each side of a regular hexagon inscribed in a circle subtends an angle of $60 °$ at the centre

In the $△ O A B$ ,

$O A = O B$ (radii of same circle).

Hence, $O A B$ is an isosceles triangle.

Hence, $∠ O B A = ∠ O A B$

We have, $∠ O A B + ∠ O B A + ∠ A O B = 180 °$ (By angle sum property)

$∠ O A B + ∠ O B A = 180 ° - ∠ A O B$

$∠ O B A = ∠ O A B = \frac{180 ° - ∠ A O B}{2} = \frac{180 ° - 60 °}{2} = 60 °$

Hence, all angles in $△ O A B$ are equal. Hence, it is an equilateral triangle. Hence, $A B =$ radius of circle.

Similarly, we can prove that $∠ O B C = ∠ O C B$ and $B C =$ radius and so on taking other triangles. Hence, all sides are equal to the radius of the circle.
Hence, proved.

Each side of a regular hexagon, inscribed in a circle subtends an angle of $60^{\circ}$ at the centre and is equals to the radius of the circle. Prove it.

Important Questions on Circle

Learn and Prepare for any exam you want !

Each side of a regular hexagon, inscribed in a circle subtends an angle of 60∘ at the centre and is equals to the radius of the circle. Prove it.

Important Questions on Circle

Learn and Prepare for any exam you want !

Each side of a regular hexagon, inscribed in a circle subtends an angle of $60^{\circ}$ at the centre and is equals to the radius of the circle. Prove it.