Question

The given diagram shows a square inscribed in a circle with center O The radius of the circle is 6 cm the perimeter of the shaded region can be written as $(p\pi + q)$ centimeters then the value of p and q respectively is __

$
  {\text{A}}{\text{. }}12,3 \\
  {\text{B}}{\text{. }}3,12 \\
  {\text{C}}{\text{. }}3,2 \\
  {\text{D}}{\text{. }}2,3 \\
 $

Answer 1

Hint:- Here we go through by finding the arc of sector by the formula $\dfrac{\theta }{{{{360}^0}}} \times 2\pi r$ and the radius of the circle are given now add the outers sides of the shaded region to find perimeter and compare it to the given answer to find out the unknown terms.

Complete step-by-step answer:

Here in the question it is given that a square inscribed in a circle with center O The radius of the circle is 6 cm.
Now for finding the perimeter of the shaded region first we have to find out the arc length that is in the shaded region.
For finding the arc length of a sector we will know the formula of arc length of a sector I.e. $\dfrac{\theta }{{{{360}^0}}} \times 2\pi r$.
And here in this figure the arc is made by the quadrant of the square and we know that the angle made by the center of the square by its two diagonal is ${90^0}$. Therefore the angle of the sector becomes ${90^0}$.
And the radius of the circle is given as 6cm.
Now put these data in the formula we get,
Arc length of a sector=$\dfrac{{{{90}^0}}}{{{{360}^0}}} \times 2\pi \times 6cm = 3\pi cm$.
Now for the perimeter of the shaded region we have to add the two radius and one arc length that is given to$(p\pi + q)$.
$
   \Rightarrow \left( {6 + 6 + 3\pi } \right)cm = \left( {p\pi + q} \right)cm \\
   \Rightarrow \left( {3\pi + 12} \right)cm = \left( {p\pi + q} \right)cm \\
 $
Now by comparing the terms we will get to know that p=3 and q=12.
Hence option B is the correct answer.

Note: - Whenever we face such a type of a question the key concept for solving the question is to first find out the unknown terms which we have to use in finding the perimeter here we don’t know the length of the arc so we first find out the arc length. Then we will find out the perimeter and compare our answer with the given answer to find out the unknown term in the given answer.