Question

What is the ratio of radii of the circumcircle and the incircle of a polygon of 12 sides
(A) The polygon is a regular one of side 8 cm.
(B) One angle measures ${150}^{\circ }$

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient
• C. Each statement alone is sufficient
• D. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Answer 1

The correct option is A Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient

Incircle radius of a regular polygon with n sides = $\frac{s}{2\times tan\left(\frac{180}{n}\right)}$

where s = length of the side of a regular polygon

n = number of sides of the polygon

All sides of a regular polygon are equal in length. If you know the length of one of the sides, the radius is given by the formula

R = $\frac{s}{2\times sin\left(\frac{180}{n}\right)}$

where s = length of the side of a regular polygon

n = number of sides of the polygon

Since we know s and n from statement 1 only, hence we can find the required ratio based on statement 1 alone.

Hence, Option (a)