Assertion (A): The length of the side of a regular hexagon is equal to the radius of its circumcircle.

Reason (R):

∠AOB = $\dfrac{360°}{6} = 60°$
∠OAB = ∠OBA = $\dfrac{180° - 60°}{2} = 60°$
⇒ OAB is an equilateral triangle
So, AB = OA = radius (r) of the circumcircle

1. A is true, R is false.
2. A is false, R is true.
3. Both A and R are true.
4. Both A and R are false.

Both A and R are true.

Explanation

In a regular hexagon, all sides and angles are equal.

The circumcircle is the circle that passes through all six vertices of the hexagon. The center of the circumcircle is the same as the center of the hexagon.

A regular hexagon can be divided into six equilateral triangles by drawing lines from the center of the hexagon to its vertices. These lines are the radii of the circumcircle.

Each of these triangles is equilateral because the three sides of each triangle are the radius of the circumcircle (two sides) and the side of the hexagon (third side).

In a regular hexagon, the vertices are evenly spaced along the circle, ensuring all triangles are congruent and equilateral.

In an equilateral triangle, all three sides are equal. Thus, the length of the side of the hexagon is equal to the radius of the circumcircle.

∴ Assertion (A) is true.

∠ AOB = ∠ BOC = ∠ COD = ∠ DOE = ∠ EOF = ∠ FOA

So, ∠ AOB = $\dfrac{360°}{6}$ = 60°

In Δ AOB,

AO = OB

⇒ ∠ OAB = ∠ OBA

Let ∠ OAB = x°

As we know that sum of all angles of triangle is 180°.

So, ∠ OAB + ∠ OBA + ∠ AOB = 180°

⇒ x° + x° + 60° = 180°

⇒ 2x° + 60° = 180°

⇒ 2x° = 180° - 60°

⇒ 2x° = 120°

⇒ x° = $\dfrac{120°}{2}$

⇒ x° = 60°

So, ∠ OAB = ∠ OBA = ∠ BOA = 60°

∴ AOB is a equilateral triangle.

∴ Reason (R) is true.

Hence, both Assertion (A) and Reason (R) are true.