Question

In the figure given below O is the centre of the circle and OPQR is a rectangle A is a point on PO such that $AO = \dfrac{1}{3}PO$ and B is the mid-point of OR. Find the area of the shaded region, if PA = 8 cm and BR = 4cm (Use $\pi = 3.14$).

(A). 132.68 $cm^2$
(B). 121.12 $cm^2$
(C). 108.56 $cm^2$
(D). None of these

Answer 1

Hint – From the given fig we can clearly make out that the area of the shaded region = area of OPQR ( (rectangle) + $\dfrac{3}{4}$(area of circle) - $\dfrac{1}{4}$(area of circle). Now use corresponding formulas.

Complete step-by-step answer:
We have been given in the question that-
AO = 1/3 PO and B is the mid-point of OR,
And PA = 8 cm and BR = 4 cm.
Therefore, we can say that-
PO = PA + AO-(1)
Using, AO = 1/3 PO in the equation (1).
$
  PO = PA + \dfrac{{PO}}{3} \\
   \Rightarrow \dfrac{2}{3}PO = PA \\
$
Now, we know the value of PA is 8 cm.
Therefore, $
   \Rightarrow \dfrac{2}{3}PO = 8 \\
   \Rightarrow PO = 12cm \\
 $.
Using the given figure, OR = OB + BR,
And since B is the mid-point of OR, therefore, OB = BR.
Hence, OR = BR + BR = 2BR.
So, OR = (2)(4) = 8 cm.
Now, the area of the shaded region can be found out by using,
 the area of the shaded region, A = area of OPQR ( (rectangle) + $\dfrac{3}{4}$(area of circle) - $\dfrac{1}{4}$(area of circle).
$
  A = 12 \times 8 + \dfrac{1}{2}\pi {4^2} \\
  A = 96 + 3.14 \times 8 \\
  A = 121.12c{m^2} \\
$
Therefore, the correct option is B.

Note – Whenever these types of questions appear, then always write down the length of the sides given in the question, and all the values, and then by using the formula given in the hint, find the value of the shaded region. We can also calculate the area of the shaded region as the area of rectangle + area of circle - twice the area of quarter of a circle (because the non-shaded area is being taken twice).